This requires you to make a kill using either a grenade or rocket launcher. All classes can carry grenades but only the heavy can use a rocket launcher. This should come naturally as you progress through the game but a good strategy is to use grenades and rockets on weaker enemies like Sectoids and Thin Men, as these tend to group up, or to finish off enemies who have low health. Once the 50th alien is killed the achievement will unlock. This carries across all playthroughs and does not have to be done in a single game.
TIP: Although you can only carry a limited number per mission, grenades are infinite, so it’s a good idea to use up each soldier’s grenade supply each mission.