In order to get this achievement, you must be at 10% health/armor which is 52.5 health. Health on the mech is shown at the bottom middle of your screen with the number 525 (representing remaining health) next to it. This means that you need to get your
mech's health at or below 52 health. You can do this by using the first mech you are given in the game and shooting yourself with the rockets. The only negative with using the rockets is that this will bring you to a limited health, making it a bit harder. If you shot directly at your feet, you will get to 25 health, which is what I did. This is however, much simpler then getting shot by enemies and then relying on luck to run away and wait the 15 secs
This cannot be done in training, it has to be regular multiplayer or Co-Op team deathmatch, which may be a bit simpler since you are playing against A.I (thanks to Dalfiuss for the info). So join a game and when you spawn in shoot yourself and get to 25 health/armor. From there, you must wait 15 secs (achievement purpose) and then just kinda chill back and pick off an enemy that a teammate may be shooting at (ha sorry teammate). I got this my first try but it may take you a few times, but you will eventually get lucky and get it.
If you downvote, please let me know why so I can improve and if I helped go ahead and slap that upvote button. Good luck out there guys!
Hoped I helped you.