It's not a very optimal build choice so I would stay away from it personally. But if one of my players wanted to do this I would encourage a Captain America build using Human Fighter and going Battlemaster. The reason for human is you want both Tavern Brawler and Shield Master feats as soon as you can get them.
You will do only d4+str damage on your main attack, but your AC will still be good because of shield and the bonus action shield bash you can use to knock a target prone every round hopefully giving advantage to your own punch and hopefully some allies attacks on him aswell. Spice up all this bare fisted punching with cool maneuvers from battlemaster and a free hand to grab, use improvised weapons, pick up disarmed weapons, and other creative things and you have the makings of an effective and fun character.
The main issue is your lowish damage, if you don't do damage and therfore are not much of a threat you are also not much of a tank as the enemies will ignore you. It might be fun to try and come up with cool stuff to do to make up for lack of a weapon, but in the end your best option is just to pick up a weapon and use it.
Talk to your DM and maybe he will place interesting and good damaging props to be used as improvised weapons around the encounter areas, or if you just don't want to use forged weapons carry around a collection of improvised weapons on you like Casey Jones from the TMNT.
But if your goal is a big bare knuckle boxer, without weapon or shield I am afraid you would need to go beyond the book and work on making your own fighter or barbarian subclass.