'Last Man Standing' is the game variant in which you must simply kill off all other players by shooting at them. If you are playing the regular versus mode, it is the mini-game with the pistol icon.
If you play this game mode enough times, you should get this achievement naturally.
However, if you are having issues, you can make it easier by subbing in human players and let your main account win.
Also, if you don't have extra controllers, you can make a custom match and turn the difficulty all the way down to the easiest setting.
Either way, make sure there are at least 3 players in the match for the achievement to unlock.