Chivalry 2

You can get fist out two ways. One: throw your weapons by pressing both bumpers at the same time. Two: hold right on the dpad and select fist. Once your fist are out, the same buttons to attack are used (RT, LB, RB) and LT to Block. Each hit will only do about 50-60 damage so you will need to hit a good amount of times to kill an enemy. Best advice is to pick someone off that's alone and not near other enemies of just join in a battle with teammates and hope you get the final hit. If someone goes down, you still have to deliver the final hit. If someone is crawling punch them and claim that kill.

As noted by jcrankin: You can also do this at the end of the game when the one team is disarmed. They count towards the achievement even though the game is "ended". I just throw my weapons and fist fight with the remaining enemies. Seems less frustrating than trying to kill people with weapons.

I found an extremely way to do this was the Coxwell map especially in the 40 player mode as it has many more AI. For this method you need to be on the Red team (Mason) and all throughout the level you will often see peasants carrying either pickaxes or pitchforks. These characters are significant because they don't block much and they only take two punches to kill so you will want to try and loop around your team and your enemies to get to these and punch them. Now how to do this you need to press and together to throw your weapons. You want to do this until you're down to just your fist and then you can simply spam to attack with your fists.

Additional ways of getting fist kills is throwing your weapons to damage your enemies and then attempting to fist fight them or simply getting lucky with unarmed kills. Also at the end of matches the losing team is unarmed for a short time so if you are on the winning team you can have an easy fist fight with these players as they can still be killed.

Really easy

From the main menu

Play > Offline > Offline practice

Map type - Rudhelm Feast

All other settings leave as they are.

When you get into the game throw whatever is in your hand and then just go and punch everyone in sight until you kill them should take around 2 min to finish.