Heavy Gunners are minibosses that appear in co-op. They may also be in Special Ops missions but those can be pretty brutal even on easy so I'd suggest sticking with co-op for this achievement. In order for it to count toward the achievement you have to be the one to deal the killing blow.
Best class to kill them as is a medic. The shotgun does a lot of damage when you shoot them in their weak spot on their back with it, and in addition the medic's defibrillator can damage them regardless of where you hit the Heavy Gunner with it.
You can track how many Heavy Gunner kills you have by going to Progress > Challenges > Badges. There's a badge that tracks each difficulty, Outgunned for easy, Gun Runner for normal, and Gun Master for hard. It just needs to be five all together, not on one difficulty, as I had four kills on normal and one on easy when I got the achievement.
Once you kill your fifth Heavy Gunner the achievement will pop at the end of that match.