You need to kill a Player Demon with every weapon at least once, which means with each of your 7 'normal' weapons and with the BFG. You can get the BFG as your third choosable item, so the match has to be at least 4 rounds long. It also doesn't matter how much damage you do as long as the killing blow was done with the particular weapon. By the way, the BFG does only damage to Player Demons if you hit them directly.