This achievement is easier than it sounds: just play multiplayer for a few hours. I got this achievement after just one day of multiplayer play (approximately 10-12 complete matches). I highly recommend playing against Geth if you want to grind this one quickly, because the Reapers are too difficult and Cerberus aren't numerous enough.
You can easily supplement this number with singleplayer kills, as they add up to a combined total, but to get this quickly, play multiplayer a lot.