There are two Fight Club locations: Ultor Dome and Stilwater Prison. Do the Ultor Dome series first, since you increase your melee damage after the third and sixth fights.
Play through the Sons of Samedi missions until you unlock their fighting style, and then go to the plastic surgeon and upgrade to the new fighting style. The kick move (left trigger) is a head-high kick that can knock down multiple opponents; charge it up and let fly when enemy fighters get close.
Before starting Fight Club, stock up on $100 hamburgers from one of the fast-food places so you can replentish your health during a fight. Spectators at Fight Clubs sometimes toss food into the ring but it can be hard to reach without getting hit.
Once the fights start, stay on the outer edges of the ring and keep moving at all times. When an enemy gets close, hit them with the Samedi kick to knock them down.
If you find yourself low on health, run to a clear area and quickly eat a hamburger; whatever you do, don't let yourself get surrounded by enemies. Once again, stay on the outer edge of the ring as much as you can.
When an enemy has been knocked down you can finish them by pressing "Y" and then following the prompts; wait until their health is really low before doing this, because sometimes they can counter your finisher. The closer an enemy is to death, the less button-mashing you have to do to kill them.
Don't do all the fights at once: if you're low on food, go buy some more and then come back.
After finishing the fights at the Ultor Dome, the fights at the prison are easier because your melee attacks are now 30% more damaging. Clearing the fights at the prison also unlocks a homie that you can call on your cellphone.