This is done in the Turkey Hunt game mode.
As soon as you start, do not fire any rounds and let the turkeys get away.
Once they are out of play, you will get a message that you failed and a monkey will poke his head out of the bushes and laugh at you. (the bushes off to the right-hand side of the screen)
For the achievement, you must run over and shoot the monkey.
Keep in mind that you must not hit any of the turkeys in order to count as a 'fail' and if you used all 3 of your shots in the round, you won't have any ammo to shoot the monkey.