Weapons and grenade mods in this game have different elemental types. You can tell what they are to the right of the stats on the gun. These can be bought in the store or looted from enemies and chests. Killing the enemies with the specific type of elemental weapon will count towards this achievement. You can check your progress on each elemental type in the challenges menu.
The rocket on your vehicle counts as an explosive weapon and will count towards this achievement. Adding elemental artifacts to your action skill will also count toward these achievements. -Thanks to KFZ Scrubs