You need to play a mission with level 20 or higher Grineer enemies. I recommend Saturn> Telesto> Exterminate. I also recommend playing solo, so you can pay closer attention for the waypoint. Play normally until you see the waypoint indicating that a Kuva Larvling has spawned. Chase it down, before it can escape. Hurt the target with normal attacks, then press X to execute it with its health depleted. Achievement should pop immediately. Video shows me doing it.
Some have reported that there is a hidden requirement of killing enemies quickly. I can't confirm, but you should play quickly, just to be safe.