This isn't as hard as it sounds.
Firstly you will need to start up a Quick Match game (from the main menu select Combat Zone then Quick Match).
Select your match type (most likely to find a match in Team Deathmatch) and search for a game.
Once the game starts go out and kill someone. When they die they will drop their weapon.
Stand over the weapon and press the required button to pick it up (Q using the default controls).
Keep hold of this weapon and hunt down the person you have just killed.
Kill them again with the weapon you have just picked up and the achievement will unlock!
Please note: If you die after picking up their weapon you will respawn with your default weapon. This means you will have to go out and kill someone else to get their weapon before trying this again.