Is there a build that focuses only on simple weapons? Are there any feats or skills that will boost simple weapons?

I assume you’re talking about a melee character, since, technically a wizard is (most of the time) going to be a simple weapon build. 

All the damage type feats: piercer, crusher, slasher would work with simple weapons. Lots of feats would. Sentinel will work just as well with a short sword as it will a great sword. Sharpshooter works with a sling or a longbow. Crossbow expert works with a light or heavy crossbow. The only thing that comes to mind that wouldn’t is great weapon master, since I don’t think there are any heavy, simple weapons. (There may be a simple heavy, or other feats I’m forgetting)
Most rogue builds will work well with a simple weapon, since the weapon is mostly just a vehicle for sneak attack — particularly at higher levels, where if you’re roll 6 sneak attack die, it won’t matter much if your base weapon damage is a d6 or a d8, or a d4 for that matter. 
Really most builds can work. I’m running a dexadin right now with a short sword, and he’s quite effective. He’d be a tiny bit better with a rapier, but not so I’d notice. 
That said, a barbarian might notice the difference when their brutal critical die are lower, so the difference in damage would be magnified. 

But I’d say any kind of character could work, it won’t be optimized for damage, but it should still be effective.

There aren't any feats that give bonuses exclusively to simple weapons.

Simple weapons are by their nature not as good as equivalent martial weapons (with a few exceptions like the spear, which is better than the trident by virtue of being completely identical in stats but working with the Polearm Master feat).

If you're looking for a build that specializes in using simple weapons and is oriented more toward combat than spellcasting, the only ones that come to mind are the Armorer Artificer and the Monk, and that's because both classes have special circumstances that make them better.

Well, there are some Ranger builds I've seen that are based on getting access to Shillelagh and thus use either a club or staff.

Hmmm interesting, I'll keep these in mind, I was thinking of making a build around the lizard race and their ability to make simple weapons out of plants and animals/monstrosities.

It occurs to me that you could theoretically build a character around throwing javelins.

I thought about it but, the damage output is just really low. It could be fun though. There is a few perks that can help but it's bottom tier when it comes to damage even with hex blade or eldritch knight. Maybe psi knight can make it work 

Yeah, it's hardly going to be a great build, it's just the only thing I could think of that could be considered a functional build around a simple weapon.

a Light Crossbow is a simple ranged weapon, plenty of ranged classes wouldn't' mind building around that,

many rogues go for the double dagger build.

Pick up Shillelagh via any of numerous means (Druidic Combat style, dipping a druid level, druidic magic adept) and go ham.  Bumps any big stick up to 1d8 magic stick and lets you use WIS/INT/CHA instead of STR if you wish.

You can blow a concentration prior on any number of things (magic or elemental weapon, various smites, hex or hunter's mark) that enhance the stick, and then stack booming blade or green-flames on top of that when you attack with your 1 attack.  Quarterstaff is also a monk weapon, and works with pole-arm expert feat.  If you go with clubs, they're light and can be dual-wielded fairly easily - and easily picked up from the environment in situations where you're bereft of weaponry.

Advantage of going this route is that both STR and (to a lesser extent DEX unless you need it for AC) happily become dump stats, and there's no need to twist your build looking for Martial Weapons.  You may need to twist it other ways to get some cantrips, though.

Quote from BubbaGrump >>

Pick up Shillelagh via any of numerous means (Druidic Combat style, dipping a druid level, druidic magic adept) and go ham.  Bumps any big stick up to 1d8 magic stick and lets you use WIS/INT/CHA instead of STR if you wish.

 

 I don't know of any way that Shillelagh can be used with INT. There's a couple of ways one can use it with CHA (the ones that come to mind are Pact of the Tome and Bard Magic Secrets), but I don't know of any INT caster features that would allow Shillelagh to be added to their list.

And this is increasingly turning into a build for a caster, except that casters are almost always better off actually casting than making weapon attacks.

I caught that also but just left instead of another edit.  Ostensibly if you are stacking your mental stats, then you're still likely putting WIS over STR (even if it's not your primary stat), which is what it would default to in most cases where you pick it up.  Monks, Druids, and Clerics would all be going for WIS.  Locks and Bards would be ok with CHA.  I wouldn't even attempt making a melee wizard or non-dragon sorcerer - way too squishy with their HD.  Every other class would/should be universally going after a martial or finesse option, rendering it moot.  The edge case would be War Wizards, which are really boned unless their race picks up the slack on weapon options.

I thought about it but, the damage output is just really low. It could be fun though. There is a few perks that can help but it's bottom tier when it comes to damage even with hex blade or eldritch knight. Maybe psi knight can make it work

It kind of makes sense when you think about it. The whole point of martial weapons is to do what simple weapons do, but better. You can add a few things to buff up a staff or javelin or something, but the baseline is always going to be lower because they are intended to be worse.

5e is consciously avoiding the 3.5/4e phenomenon of adding a bunch of feats and specialist classes to make every weapon potentially great. On one hand, I miss the opportunity to tease out all the features I could find to optimize something that would normally be lousy, but I don't miss scrolling through literally thousands of feats.