(MatthewMk2 isn't wrong but this is an expanded explanation)
In this game, "obliterate" means to gib the corpse. You can do this with any weapon, even your fists (which you need to do for another ahcievment, anyhow). Simply kill an enemy, and if it isn't already gibbed into pink mist (which can happen, say if you charge your shotgun and hit a weaker enemy point blank), then keep shooting the corpse until explodes into a glorious bloody cloud of gore.