I think it would be really cool to make a build focusing around the vampiric bite that Dhampirs get as unlike other natural attacks other races get, it counts as a simple weapon and scales from Con which is neat. My only ideas are maybe something with the paladin to smite or maybe a eldritch knight so you only have to focus on intelligence and con. I know that using the bite for damage is very suboptimal compared to other choices but I think its super cool so I would appreciate any help on this. 

The best use for the bite I can think of is on a rogue, since applying a huge additive buff to 1 ability check or attack roll is 100% a rogue's jam. 

If only you could sneak attack with the bite. Rogue works, but more because the class is super happy to pick up that always on spider climb.

The first thing you have to consider if you want to pursue a Dhampir bite build is that you will run into a magic weapon problem once resistances start becoming more prevalent. You can technically turn your Bite into a magic weapon by becoming a Kensei Monk, but trust me, that build path is a mess. Instead, you'll need the Insignia of Claws. It's an uncommon magic item I believe. Getting your DM to be on board with making one available at some point is key.

The two classes I like most for this approach are Paladin and Ranger. The half-casters both have ways to buff the bite attack to deal more respectable damage, but also have something else in common. They have good concentration spells. Being able to pick up resilient (con) as a way to boost concentration saves, hp, and the damage of your attack is super elegant. The Paladin is better for burst damage and makes a better tank, because it's a freakin paladin, but the Ranger is really interesting because it offers a number of damage boosts that work with the healing clause. See, you only heal equal to the piercing damage dealt. Smite won't work, but Hunter's Mark and the Hunter Ranger's level 3 Colossus Slayer ability both add damage based on the type of the attack. You won't be dealing as much damage, but you'll be much tankier than your average ranger.

I never thought about a ranger at all, that could be really cool. What do you think of a fighter? With their access to additional attacks and ASI's i think it would be helpful in trying to make the most of the bite. I'll look more into ranger as well to see what they have to offer 

There are some really nice synergies with fighter, but the class as a whole has a couple problems when it comes to being the base of a bite build. First, I think the ability to use resilient (con) to boost your attack stat is actually a gigantic part of what makes it strong. Fighter negates that benefit by providing constitution saves already and doesn't take great advantage of the boost in concentration saves either thanks to the Eldritch Knight being the only caster and a 1/3 caster at that. The other part of it is that the bite doesn't mesh well with how the Figher boosts damage. The fighter is all about taking the most attacks possible through action surge and extra, extra attack. Making a million attacks is much less meaningful when they're all working off of a d4 bite instead of something like a 2d6 greatsword. The bite wants things that boost the damage of each bite, like Hunter's Mark. I think the fighter makes for a much better dip to slap on paladin or ranger base with this concept.

Super thank you for the well thought out reply I didn't even think about that. 

How would you suggest I build a Dhampir ranger? The subclasses that I have access to are the Hunter, Swarmkeeper, Fey Wanderer, and Beast Master. I haven't looked too deeply into all the options but it seems as though the Hunter might be the best option? With its free 1d8 damage from Colossus Slayer.

Ey no problem Mail. 

Hunter is cool if you're trying to maximize the healing bite, which is definitely an interesting angle. But for my money I'd go Swarmkeeper. Dhampir with a bat swarm? Shit writes itself. 

Using point buy you can get a starting spread of 8/14/15/8/15/8 with 2 points left over to make any one of strength, intelligence or charisma a 10 based on preference. Go for a +2 con/+1 wis boost from Dhampir. 14 dex sets you up perfectly for medium armor + shield, 17 con sets up resilient, and 16 wisdom keeps casting relevant. I wouldn't dip and just take swarmkeeper all the way to 11 if you even get that far.

1. Deft Explorer, Canny [Stealth], 

2. Fighting Style: Druidic Warrior [shillelagh, thorn whip]

4. Resilient (con)

8. Wisdom + 2

Shillelagh, Thorn Whip and a workable wisdom means there are some good alternatives to biting things. Thorn Whip is super fun with Spider Climb to try and set up situations where you pull enemies off the ground only to fall prone and take extra damage. 

I like the idea of being more of a controller Ranger with this, using spells like entangle, spike growth, plant growth and conjure animals. Hunter's Mark is always available for more damage, but since building around the bite isn't really about great damage to begin with why not lean into other methods of influencing the battle.

You don't need to sneak attack with the bite to use it, and I would expect any favorable ranger build for a dhampir to be worse than a paladin one, because paladins are more melee-focused - if you lean into the bite as a melee weapon.  If you use it the way I suggested for a rogue - biting something else as prep for a big attack - then it could work very well for a ranger.

The core of how I'd build that is spells like [Tooltip Not Found] - bonus action cast the spell, first attack eat a bug from a jar of crickets you carry, second attack take the actual shot.  I agree with Zot that Swarmkeeper is the best subclass for this of the ones you can pick from - Swarmkeepers are all about one shot that does well, and not for nothing they are quite comfortable being in melee.  Hunter's Mark is also useful here for ability checks - Mark your cricket before eating it, and your check bonus goes up by 1d6.  Your Swarm's 1d6 also stacks.

But if you want to go really big on the one shot, I'd make a Rogue - an Arcane Trickster or a Phantom, since you have access to Tasha's but not Xanathar's.  Without access to Extra Attack, the primary use of your jar of snacks is for ability checks, but if you can prep for a battle - as Rogues and some Rangers are very good at - you can use it to buff your opening shot, which is your most important shot.  Since you only get it pb/long rest, this seems ideal.  It is unfortunate you can't Sneak Attack with the bite, but Rogues are the class most invested in landing a single hit, which is the thing the bite is best at.

I avoid using "bag of rats" strategies so the Rogue is a much less appealing approach for a Bite-centric build to me. Even with the sacrificial fodder, it just doesn't have the same feel. That character isn't draining the life out of their foes, they're sucking juice out of bug batteries so they can stab and shoot better. I think Dhampir make excellent Rogues, but not for a character looking to bite actual enemies in combat all of the time.

Paladin isn't better or worse than ranger for this concept it's just different. Different set of skills, different tactics. 

... aaand now I'm making a Lizardfolk Dhampir  Phantom Rogue / Gloomstalker Ranger

Hmmm.

Vampiric Bite

Your fanged bite is a natural weapon, which counts as a simple melee weapon with which you are proficient. You add your Constitution modifier, instead of your Strength modifier, to the attack and damage rolls when you attack with this bite. It deals 1d4 piercing damage on a hit. While you are missing half or more of your hit points, you have advantage on attack rolls you make with this bite.

When you attack with this bite and hit a creature that isn’t a Construct or an Undead, you can empower yourself in one of the following ways of your choice:

• You regain hit points equal to the piercing damage dealt by the bite.
• You gain a bonus to the next ability check or attack roll you make; the bonus equals the piercing damage dealt by the bite.

You can empower yourself with this bite a number of times equal to your proficiency bonus, and you regain all expended uses when you finish a long rest.

To break this down, the bite has the following properties that are relevant to building around it:

1. It has no weapon properties to exploit (e.g. Finesse for sneak attack, etc.)
2. It's a "Natural Weapon," and a "Simple Weapon," and a "Melee Weapon" but not an "Unarmed Strike." Monks, Tavern Brawlers, Lycans, and the Unarmed Fighting Style aren't going to find much traction here. 
3. It does not require any free hands to wield
4. It apparently must be Constitution based rather than Strength based, not "may". This in turn means that other features that have ways to let you use a stat instead of strength for melee attacks (Hexblade, Monk, Artificer, etc.) can't subsitute for this bite, its hard-coded to Constitution. This may not be RAI... 
5. It deals Piercing damage.
6. It provides Advantage when under half HP.
7. Proficiency bonus times per day (2-6), can "empower" it to heal an amount equal to its piercing damage (not other bonus damage), or gain a +X bonus to next attack or skill check equal to that damage. 

If we're talking about a "Bite Build," #7 isn't quite frequent enough to singlehandedly define the build... but it certainly could be part of a build that overall has a few ways to heal itself in combat, or to provide static bonuses to single huge swings. The always-on unique properties of #2-6 are more the sort of thing to really build around, rather than serve as cherry's on top. 

Con-Based Builds

The Rune Knight Fighter is one of only a handful of classes that uses Constitution to set its DC's.  However, other Fighter subclasses also use Constitution modifier in some notable ways, including Cavalier (Warding Maneuver reaction con mod times per long rest), Champion (fast healing is 5+Con Mod per round when under half HP), Echo Knight (Unleash Incarnation extra attacks con mod times per long rest, and Reclaim Potential healing too). The Barbarian's version of Unarmored Defense uses constitution modifier+dexterity modifier for AC, but often has difficulty justifying the points needed in Strength AND Dex AND Con to take full advantage. The Beast Barbarian also has con-based save DC's for its Infectious Fury, and Call of the Hunt , and the Storm Herald Barbarian's Storm Aura does as well, and the Wild Magic Barbarian's Wild Surge too.

But as discussed above, any caster that wants to focus on maintaining Concentration might also be considered a secondarily con-based build, so I suppose you can lump melee casters into this pile too. The Aberrant Dragonmark feat lets you cast a Con-based spell... but it isn't the sort of feat that lets you continue casting it with spell slots, so that's fairly limited. 

Natural Weapons

The Beast Barbarian's abilities that enhance its Natural Weapons enhance its Natural Weapons, not any others that you may already have from another source. And the Lycan Bloodhunter's abilities enhance its Unarmed Strikes, not Natural Weapons. So there really isn't much out there that keys off of the "Natural Weapon" descriptor, other than the Insignia of Claws. Though, the Moon Druid's Primal Strike would apply to this racial Natural Weapon, if you still have it in your Wild Shaped beast form (which yes, you probably do). Many Beasts have decent but not amazing Con scores, often around a +3 modifier...

Hands Free

This could be valuable for a grappler build that wants to grapple two enemies, or one enemy while holding a shield, but doesn't otherwise have enhanced Unarmed Strikes to do damage? Unlikely, however, since a grappler build would probably also be building for Strength, and not wanting to use a Con-based attack... Expertise in Athletics, and the Bite's limited use bonus to skill checks could help patch this a little bit, but the skill check bonus is too limited to use consistently for grappling round-to-round. 

Piercing

The Piercer feat is a good reliable boost to damage per round, if you don't try to save it for 1's but rather just reroll anything that's below your dice's average roll you should come out with a positive boost to DPR in the long run. The feat does unfortunately boost Strength or Dex instead of Con... but you're likely going to want Dex in this melee build anyway. 

Advantage

Rogues obviously care about always-on advantage quite a bit, since it allows Sneak Attacks... but as a non-Finesse weapon, that unfortunately is not an option with this Bite. But Crit Fishers also care about such things, so Paladins and Champions might appreciate this feature! The Elven Accuracy feat makes advantage even better than it already is... and its arguable whether a formerly-elven-dhampir is intended to be able to keep the benefit of its Elven Accuracy feat it took before adopting the lineage, or might even be allowed to take it in the future after becoming a Dhampir if their original race was elf? But Elven Accuracy only works on Dex, Int, Wis, or Cha-based attacks, and again, there is no way to change the Con-based bite to anything else RAW, though I suspect that wasn't RAI.... 

Fixing that D4

While a Monk's martial arts can't make the only-Con bite Dex-based, it can treat the bite as a Monk Weapon and enhance its damage dice. Monks don't particularly care what their Dex is, if you have other ways to provide a good attack modifier and AC, and most subclasses also don't particularly care about their supposedly-important Wisdom for anything other than Stunning Strike DC. If you can swing the 13 Wisdom and Dex to MC into and out of the class, 5 levels give you a d6 bite, and 11 give you a d8... as well as a wealth of other class features that feel good on a melee character. The Kensei's level 6 magic bites would be helpful, though you'll probably need to carry around a useless dagger or something to benefit from other Kensei class features. 

Otherwise, you're probably just looking at stacking bonus damage onto your bites to make up for d4s. Hunter's Mark is good because it boosts your piercing damage rather than introducing a new damage type, and can be easily nabbed with the Fey Touched feat. But other than that... Battlemaster maneuvers, Sword Bard flourishes, Monster Slayer/Hunter/Gloomstalker/Swarmkeeper bonus damage... that might be it? 

Conclusion

There's SO MUCH Barbarian synergy there, which is hamstrung at the last moment when you remember that Rage is overly restrictive to only apply bonuses to strength-based attacks (boooo). A couple of splash Barbarian levels for a character that intends to build around Dex and Con are still very worth it for Unarmored Defense alone, but its hard to recommend advancing far enough into any of the Barbarian subclasses to pick up any interesting subclass features with con-based uses or saving throws. Instead, the Rune Knight, Champion, Echo Knight, or Cavalier would seem to be the Con-based classes of choice for a bite specialist, or Battlemaster if you want to be bumping your d4's up with extra maneuver damage. Bumping a d4 up to d6 is only 1 more average damage per bite, and up to a d8 again only 1 more, so multiclassing monk for that alone doesn't seem important for that alone... but might still be fun for other class features? A Ranger MC into Monster Slayer, Hunter, Gloomstalker, or Swarmkeeper is probably more effective to provide a damage boost through an extra dice... but since it has the same MC requirements as Monk, maybe both!

While the Champion often gets poo-pooed as an underwhelming subclass, the unique combination of being a Con-based attack AND providing frequent advantage AND having healing-under-half-HP synergy with their T4 theme makes me think that this might finally be their chance to shine. 

Recommendation:

1. Barbarian 1, for extra HP out the gate and the foundation of your Unarmored Defense lifestyle. Start with 13 or 14 Str to allow multiclassing out, but everything else is in Con (17) and Dex (16) for a starting AC 16 and full +5 to hit bites. Go ahead and plan for 13 wisdom as well, just in case you want to MC monk a little bit for some speed enhancements, evasion, deflect missiles, etc etc.
2. MC to Fighter 1, and take Superior Technique or Blind Fighting for your style. Recall that Dueling won't work on your Bite, since it isn't wielded in a hand :(
3. Either ride Fighter for the rest of your career (picking up Piercer, Skill Expert, Tough, Shield Master, ASI's to take Dex and Con to 20, etc.), or splash in 3 levels of Ranger for some bonus damage dice and Hunter's Mark, or that and 6 levels of Kensei?

I made a build, comes online best at lv6. It's an unusual multiclass of Ranger and Warlock both (w3, r3), with a pretty hefty focus on Constitution as well (My first level was as warlock for Wis save proficiency but you can do whatever). I went hexblade warlock, and because the bite counts as a simple melee weapon, that can be your hex weapon, so I use charisma for it. Went swarm keeper ranger so I could have a swarm of bats, but also because the swarm allows you to deal an extra D6 of piercing damage on an attack you make on your turn, and I took hunter's mark to add another D6 to that piercing damage. Most of the spells I took aside from hunter's mark were just utility spells, but maybe there are other spells besides Hunter's Mark that would work well. I wanted to emphasize constitution and piercing damage heavily so I could always have a good amount of hit points, as your bite restores HP equal to the piercing damage dealt with the attack. 

Is it a great optimized build? noooo not at all you could totally get more damage out of the vampiric bite. Is it kinda rad? hell yeah

None of the features actually work the way you are using them though. Hexblade doesn't work for the bite because the ability specifies that "you can use Cha in place of Str or Dex." Since the bite isn't allowed to use Str or Dex and is required to use Con instead, it doesnt qualify to substitute the damage (this also prevents it from benefitting from Artificer or Monk benefits, though Monks can change the damage dice). Likewise, Swarmkeeper deals damage after the attack, not as part of it. It is a separate damage source, so it doesn't apply to the bite damage (though it does force casters to make a second concentration check, making Swarmkeeper an excellent mage slayer). This build is very interesting and cool, but relies entirely on a DM that lets you play loose with the rules