HAWKED

Fairly simple on the PvE aspect of the game. Anytime you see a red exploding barrel, shoot it with enemies around. Most of the little platforms/buildings around the map have a barrel with 2-3 enemies around it. Just shoot them from a distance and the achievement should pop shortly.

This damage is cumulative and doesn't need to be done at one time or in one match. Unfortunately, there's no way I saw to track progress.

Also, certain guns can cause explosions, like grenade launchers. This is how I got my final damage needed.