This quest requires you to deal damage using weapons in the sub-machine gun class. This includes all rarities of the makeshift SMG, primal SMG and mechanical SMG.

A sub-machine gun has a high rate of fire but a short range so it's most effective up close and personal.

Team rumble is a good game mode to do this in as you respawn and keep your weapons so you can rinse and repeat until you have this quest.

You can also get progression towards this quest by dealing damage to NPCs who attack you on sight.