This achievement is a lot easier than getting ten kills in a row as Lynch without being hit.
WHILST PLAYING AS LYNCH in coop mode, down an enemy. This can be done by shooting them in the foot or leg. They are downed if they are still alive on the ground. You can tell they are as they will be shouting and wriggling on the ground, they stay like that for about 30 seconds. Pull out your shotgun and cap them in the head.
I got this on the 'Withdrawal' mission. When securing the lobby as Lynch there is a guard on the top floor, I shot him until he was downed, pulled out my shoty and blew his face off.
Video is not mine, but shows you how to do it.