Hitman: Absolution - Gunslinger

Gunslinger requires you to get kills using the revolver. You can use any of the missions where 47 starts with a revolver or you can play a level with a lot of street cops (like Run for your Life). Once you have a revolver, start shooting.

The only downside is that there is no stealthy way of doing this. You have a limited number of shots before reloading and will almost certainly find yourself in a gun fight. Which is likely the point. Don't be discouraged if it takes a couple of tries.