For context, I'm starting a campaign where character creation is a lot more flexible than normal. I've already homebrewed a background for it, and I'm working on a couple of technology items for a character, since it does start out in a modern setting. This particular detail has me confused, though.

One of the PC's is a human, but his teeth are "super sharp, like deadly sharp," and he bites people as an attack (the player's an interesting guy). How should I go about that? Realistically, I'm not sure how effective it wold be, since combining the inhuman teeth with a human bite strength is kinda tricky to calculate. What kind of damage could it do, and how sharp should I allow the teeth to be before it gets unbalanced? How thick even is the monsters' and other species' skin? This has caused me to research human bites far more than I'd like, and I'm still puzzled. Any help?

A regular human bite would technically count as a normal unarmed strike just like punching or kicking, so standard Strength modifier +1 for damage. For these “super sharp” teeth I might go as high as 1d4 + Strength like a club (only slashing damage), but certainly not as high as a lizard person’s 1d6 + Str. But that’s just me. Make sense?

You can't just say you're good at something and receive a mechanical benefit for it. If you could, half of the characters in the game would be 300 year old elves who have learned every language, weapon, tool, and armor proficiency because they've lived "really long. Like SUPER long." And the other half would be dwarves who claim their skull is so thick they should get advantage on all concentration saves. You see where this kind of thing leads?

If you want a natural attack, find a legal player option with a bite or similar natural weapon attack and work backward to figure out how it can fit your character. Maybe you're a reflavored lizardfolk or tabaxi. Maybe you take the Tavern Brawler or Fighting Initiate (Unarmed Fighting) feat and persuade your DM to allow piercing damage instead of bludgeoning damage. Maybe you wield a dagger and just describe yourself as unarmed. You can get as creative with how you look as you like, but not how the character works.

edit - now realizing I'm in the homebrew forum, I'm still sticking with this answer because I don't think you should homebrew something when you can achieve the result without homebrew. And in this case I think there are a lot of ways to do that.