In order to obtain this achievement, you have to kill a boss with one of the enemy characters in that world. You need to unlock the enemy character, and then fight the boss and kill him with the unlocked character.
The easiest way to get this achievement is to play for a bit, and then go to the first circle on the map called "King's Arena". Finish the arena by killing the waves of enemies until they stop coming (there is 10 waves of enemies). After you finish the arena, you will unlock the very first enemy character. Exit the game, and start up a single player local game. Choose the newly unlocked enemy character as your character, and proceed to the "Barbarian Battle" level. Kill the boss here, and the achievement will be yours.