The Fights are available in the mercenaries' compound once you complete the Barracks faction mission and renovation. During Sequence 3, you will be presented with three missions; the Barracks mission is located in the northeast part of Rome. (The other two missions are related to the Thief and Courtesan factions.) Once you have completed the mission and have renovated the mercenaries' faction compound, you can enter and go to the basement to participate in the Fights, a hand-to-hand combat competition where you bet on yourself against 1-5 opponents.
To get this achievement, you must bet 500 florins on yourself in the hardest possible fight, you against 5 mercenaries with a 40 second time limit. These mercenaries are rather difficult, as they have a good amount of health and can throw sand in your eyes and dodge your attacks. However, you can win this pretty quickly with the following method.
When the bell sounds, immediately sprint towards the group, preferably one end of the row of combatants. Attack one combatant constantly (press the left analog stick towards that combatant), mostly throwing punches (X), but you'll also want to throw in some kicks (A) just to keep him on the defense. Keep an eye on the other combatants' health meters -- when one flashes, it means that he is preparing to attack you. Once he begins to attack, you'll want to quickly press RT and X and tilt the left analog stick in his direction to counter his attack. This counterattack will either result in you avoiding his attack and him avoiding your counter, or, in some instances, will give you a scripted combo and an instant defeat of that particular opponent. Time is of the essence in the Fights, so these instant wins are important.
Keep focused on one opponent at a time, while countering the others when they attack. If you get down to a single guy who keeps dodging, stop attacking and wait for him to bend over to throw sand. When he does, quickly attack him and keep swinging until he's down. Following this method, you should meet the target time with a few seconds to spare; I was able to win with about 5 seconds on the clock with 2 counterattack instant wins and 3 punch/kick sequences.
** Please leave constructive comments if you're going to downvote the solution. **