This achievement requires you to kill 20 enemies using any type of explosive objects.
NOTE: Grenade kills do not count, HOWEVER, an explosion of a truck or barrel resulting from a grenade does count.
You want to keep an eye out for red barrels and trucks/vehicles. These will be your "objects".
I would recommend you just play through the campaign twice without going for any kill x amount of enemies with x weapon, then on your third campaign run through, go for all the achievements you have left.
There are a number of parts in this game where a truckload of about 5 to 6 enemies will show up, as soon as you see them slow down to a stop, chuck a grenade under the truck, most of them will run to a different position, but 2 or 3 will stay behind the truck for cover.