A "first strike kill" is awarded to the first person that gets a kill in a multiplayer match.
Personally I unlocked this on a Control map. Instead of heading immediately to the capture point, start running for your opponents capture point instead (the flag at "A", "B", or "C" where the enemy team spawns). I personally nabbed in by tossing a grenade directly into the flag area and got 2 kills and this trophy.
However this will unlock on any type of multiplayer match so just be the first one to get a kill and the trophy should unlock immediately following.