First of all, to earn this achievement you're gonna need to find yourself a boosting group, the online to this is still pretty active, but the chances of finding 4 other random players who are willing to help you get an achievement is pretty slim.
Now, the fastest way to get this achievement is to play Co-op Elimination and set the mode to Firefight, from here you'll want to enter the advanced options screen, turn the helicopters off, turn the infantry off and set friendly respawns to infinite, keep the difficulty at default, set the map to Headquarters.
By turning enemy infantry/helicopters off you'll have forced the game to spawn the minimum amount of enemies, which is 20, we've set the map to Headquarters because it's one of the smallest maps and therefore quickest maps to complete the elimination goal on.
Weapons/Classes don't make much difference, I opted to use the M36 SAW, only because of it's massive clip, it meant I didn't ever need to reload, but like I said, it doesn't really make a difference what you use.
Lastly you'll want to place the Drone a little ahead of you, just to help flag up where the enemies are.
Once you've got the routine down you should be finishing each game in around a minute, sometimes a little more other times a little less, but overall you shouldn't be looking at much more then a minute per game.