Here's the technique I used to farm my 1,000 kills, which you'll also most likely get the Exterminator (7 rapid kills in a row without dying) achievement in the process with very little effort.
Game mode: Juggernaut
Level: The Bunker
Time: 5 minutes (max time)
Take the minigun back to your spawn and point it outwards where the AI will proceed to constantly walk into it.
At some point, the AI will pick up the Kamikaze power-up, which makes them invincible until they explode, which will kill you.
However, since you're in your own spawn, you should be able to respawn fast enough to stop them from taking the mini-gun away from your spawn. Repeat this a few times and the 1,000 kills will unlock quickly.
In the video, the kills for the Exterminator Achievement starts at around 2:18 and unlocks at 2:25.