TLDR: What base class could work well to balance the flavor I'm going for without giving up too much utility?

I have an idea for a character background, full of death and betrayal, that leads this character to be lawful evil (not toward the party, just big on a reverse golden rule and payback/revenge for every offence). The young character (19) is out looking for power and to make a reputation for themselves so they can go back and avenge the wrongs done to them, which is where the warlock multi-class would come in. With an eye to character development, I would start with a base class for a few levels, take 6 or 7 warlock levels, then go back to the main class. So I'm trying to figure out which base class would work well for both flavor and utility.

I've considered starting as a monk (destroyed monastery) and going the hexblade warlock route, or a Barbarian (though that seems strong, flavorwise, not quite the "weakling gains power" vibe I was hoping for), or maybe just a regular fighter.

I'm a newer player, this would be my first multi-class. Any suggestions would be greatly appreciated.

I am not too sure what sort of character you are looking for theme wise. THe background happens before the character starts adventuring so given your character is starting atage 19 are you saying the character's childhood is full of death and betrayal? If so are they lawful evil at that point?

You could be a monk whose monestary was destroyed and vow to develop your ki in order to extract revenge, you don't know how the story will develop you could want to make yourself open to your character would be of the type to make a pact with a powerful being in order to gain power to be or powerful in carrying out that revenge but you don;t know what will happen inthe few levels before that happens. If your family was destroyed you could initially join the army as a fighter, outwardly as a "new famility" but inwardly to become a trained killer.

However you could be in a monestry without any sort of combat training, and your initial reponse as your method to extract revenge is to seek power from some powerful entity.  This would mean starting as a warlock.

Mechanically a monk / warlock will be very sub-optimal especially if you are going more than a evel or two in both classes. A monk needs very high dex and wisdom while a warlock needs high cha, you also need decent con so the build is just to MAD.

A fighter-warlock can work but is probably more effective if you only take a level or two in one class and concentrate on the other.

Other classes that work well with warlock are other charisma based spell casters, (Bard, Sorcerer, or Paladin), if you plan to go paladin in partuicular I wouyld discuss things with your DM as to how your paladin Oath and  warlock pact will combine (do they work together in that your pact helps you carry out your Oath or does taking the pact tresult in you braking your Oath.

I'm going to agree with the above response.  You have a character who you want to become a Warlock with vengeance motivations.  Why does the character have to have a class prior to that?  Can't they be any number of commoner positions delineated as backgrounds?  So I'd just start as a level one Warlock, and if at level 6 or 7 you want to multi class, you'll have played enough to see a reason for the character making some breaks from his patron devotion to develop features from a new class.  If you want to play a warlock play a warlock, no need to multi class if you don't have a strong desire to play that base.  If you opt for your base and play the base, you may wind up going in an entirely different direction as you play the game.  

If you are all starting out at higher power levels so everyone's around level 8 or 9, that's an occasion to engineer a multi class balance.  But "pre-gaming" too intensely on planned level progressions and backstory-determined character arcs before the game's played risks losing sight of playing the game.  

I guess saying the character's backstory is full of death and betrayal wouldn't quite be accurate, I was thinking more of one violent defining moment when the people/system that should have provided protection instead became the oppressors, that being the reason for the character to start as lawful evil. I would play this character in a group that starts at low level, as a way to almost literally have the character grow up. I didn't want to start with a sorcerer as I felt like being born with that power would take away a lot of the motivation to enter a pact, paladins seemed too much on the good side, and bard didn't seem like the class that a young rash kid with a chip on her shoulder would choose as a path to revenge. That's what got me started looking up unique warlock multiclass builds, which I know aren't optimal but some seem to be doable with a little planning. Thats part of why I'm thinking so far ahead in character development. It also makes sense to the dabbling writer side of me that a reckless youth could mellow out and mature out of wanting revenge, which would be the reason for only taking a few warlock levels. I wouldn't be trying to force anything story/adventure wise, I'd clear all this with the dm first of course to make sure it would work in the campaign.

Paladin do not have to be good, an oath on vengeance seems to fit your concept pretty well. In fact you could play a single class vengeance paladin with that concept. 

I am also not convinced your character would be LE. Having an overwhelming desire to destroy evil doers with no hint of mercy strikes me more as LN. If a group of Elves had murdered your family, and you vow to kill every elf you can, that would be LE.

By the way 6 or 7 is a lot of levels. One very playable option I see is either before the incident or in response to it you learnt basic fighting skills and start as a fighter but agree with your dm that early on some entity will tempt you with great power to seek revenge, taking Warnock at level 2. As the story progresses you can then decide whether to continue gaining powers from your pact or from martial fighter training. If the later do you try to break your pact (for which your dm can decide the conseqences) or continue to stick with it.

Mechanically this means you get heavy armour and if you take take a pact of the blade can attack with charisma needing you only need 15 strength (not even that for a dwarf). For an optimal character you would want to either switch back to fighter after 2 or 3 warlock levels or stick with warlock but switching back to fighter after 6 or 7 warlock levels should be playable.

I don't know about how well this would fit into your backstory, but if you're going hexblade it could pair well with a Rogue character, since you can still land sneak attacks with a charisma-based Hexblade attack, as long as your weapon is finesse or ranged. The option that fits best, mechanically, would probably be Swashbuckler Rogue, since that class also gets a lot of benefits from a high charisma (for example, you add your charisma modifier to your initiative bonus). Mastermind is another good choice... it has a lot of skills that benefit from a high charisma, but doesn't have as many obvious benefits as Swashbuckler does.

It would help to state what your first priority is here. Is it to play a MC warlock or is it to fit your story?

If it's just about the story, then I've got to agree with MidnightPlat that you should jut start out as warlock and play out your growth organically through the game. Part of the fun of D&D is that you can change the story and the story can change you. Any time I've had a broad plan for my character over the course of a campaign, I've ended up being swept into the story, my character gained a life of their own through interacting with the world and the other characters, and my initial plans kind of fell by the wayside as other things became more important and more fun to pursue. In short, come to the table knowing who you are and who you were, but not necessarily who you are going to be. Writing your story ahead of time deprives you of the chance to play it out.

The other thing I think that's being conflated here is that warlock = evil/revenge/edginess and other classes don't, so the way to depict a character going evil is by giving them warlock levels and the way to redeem them is to take levels of something else. This line of thinking is just really limiting you. Any class can be played as someone seeking power for evil purposes. As mentioned above, Oath of Vengeance seems very much in line with your character concept. Fighters, monks, wizards can all be pursuing their power for evil, and warlocks can be played as selflessly endangering themselves to help others. It's not what you can do, it's what you do with it that determines your alignment. So play the class that sounds fun for you - the more you enjoy your character the more you will get into their head and the more it will take on a life of its own.

On the other hand, if you really just want to play a warlock multiclassed with something else, I'd second Trans and recommend rogue. Even once you give up your warlock ways, you'll still have all those powers - and many of them compliment the most common/most effective rogue playstyles.

Thank you for those ideas, and your advice. I'll look more into the vengeance oath paladin and the rogue, they both sound like good fits!