This quest requires you to eliminate enemies with common or uncommon weapons. These are they grey and green coloured weapons. You can use any type of weapon to get this quest.

This quest should be straight forward but I'll add some pointers to help out.

You will be more likely to win a battle using common or uncommon weapons nearer the beginning of the game before enemy players find decent loot. If you are in the last 10 and try and fight every battle with a common or uncommon weapon you will likely be eliminated because they will be using stronger weapons.

Team rumble is a good game mode for these types of quests. If you drop in a load yourself up with uncommon weapons you can get several eliminations each game by dropping in close to the enemy and eliminating them before they can react. You will respawn and keep you weapons so you can rinse and repeat until the game is over.

You could also keep an uncommon weapon in your inventory and switch to it when you target is at low health and finish them off with your common or uncommon weapon.