So I wanted to start a boxing RP as there aren't really many out there. My idea was that you start as an amateur boxer, work your way up fighting for any titles you can until old enough to box professionally. Then, you box in the professional game etc.

I would really like to do this but not sure how the fights/training would be simulated. Any ideas?

Contenders system.

Fight might also work, or the old Street Fighter game (though that might be a touch ridiculous). There are also a shocking number of wrestling games (like, more than 5). I think you could also run it in a number of other games, but these are more focused on the fight and thus may work better.

GURPS, with all of the combat rules used.  Lots of ability for customizing your character, and having detailed fights.