For this trophy you are required to have the pickaxe for Bardin, which can be acquired by playing the Karak Azgaraz DLC maps, Khazid Kro, Cursed Rune and Chain of Fire. Any color rarity will do, however mine was achieved by using an orange tier pick.
Once you have the pick, you need to find a Stormvermin patrol, not a single Stormvermin, as apparently the trophy is glitched slightly. Once you find a patrol, you need to kill one of them with just one blow without any of your teammates doing any damage to your target. In order to do that you need to use the charged attack, you should hear a thud sound when it hits if you have charged it sufficiently. Holding the attack for about 2 seconds should suffice, however, waiting for a patrol to pass you first and then attacking them from behind with a charged attack will make this much easier. It does not need to be a headshot, you only need to kill a Stormvermin from a patrol in one hit without any other teammate damaging them. Difficulty does not matter other than how many Stormvermin are in a patrol, higher difficulties mean more tries attempts per patrol but also means enemies will hit harder.