This achievement requires you to kill 1000 zombies in the menu mini-game. The good news is the kills don't have to be in a single game (you can die and start over, the count will be cumulative). The bad news is that if you quit out of the game completely and restart it, it will reset the count to zero.
See the achievement for killing 5000 zombies for the full instructions:
You can get this one in about 20 minutes of play.