My Coop Partner Breddist and I did this relatively simple on heroic without any skulls on
You can watch the Videos of our 4-Part Walkthrough below for a complete guide.
The first two Elites are really easy to miss: they are across the big hole leading to the pool of green goo at the bottom - take a plasma pistol and fire off charged shots at the Elites on the other side (if done in Coop, your Partner can protect you from any flood Enemies spawning behind you).
You know they're dead when any remaining flood on the other side start coming towards you and no more shots are fired
After jumping done, just clear out all Elites you can find until you come to the gravity lift.
Once within the ship heck every nook and cranny for elites. The red and blue Elites are the ones counting toward the Achievement - I'm not sure about the Elites in black armor, because they keep spawning at the very end of the level. Run up and down the different Levels of the hangar bay shortly before leaving with the banshees and clear any black Elites you can find - they should stop spawning after a while
Also do this on heroic, because on legendary the black Elites are significantly more
- hence my belief that they don't actually count towards the Achivement.
Here are the 4 Videos - sorry for the non-HD quality, they were uploaded directly from Twitch:
Hope the Vids can help and you can "catch'em all" - good luck ;)
(Sorry for any typos or other mistakes - this is my first written solution ever :P)