In Fight Club, you are pitted in a ring with several other people, and must kill them all. You cannot use any of your weapons, just your fists, but objects will spawn in the ring that you can pick up and beat your opponents with. I actually wouldn't recommend using any object. Even though you can knock down several enemies at once, the damage done to them is minimal and you become a target to everyone else. I would recommend throwing the object out of the ring or to the edge so the A.I. opponents can't use it.
The best way to play this is to get each enemy alone and fight them one-on-one. Use and to punch, and hold both to block. If you can land 3 successful punches in a row on an enemy, you will perform a finishing move that will greatly reduce their health. 2 finishing moves damages an enemy enough to allow you to finish them. Go up to them and hit to finish them. A short button-mashing sequence will pop up. Fill up the meter to break the person's neck and kill them.
There are two Fight Club activities with 6 levels apiece. I recommend doing this in co-op so you can work towards the "Confidence Men" achievement as well as "Crime Lord".