From the starting screen select CAMPAIGN
Select and start a NEW GAME +
Pick a save slot and on the lobby screen of CAMPAIGN +,
you will see a tab labelled as
NEW GAME + SETTINGS
SKINS, MUTATORS, AND MORE
Select that and move down to CUSTOMIZATION
From here Turn on ALL WEAPONS USE MULTIPLAYER SKINS.
Play any chapter with enemies in it, and kill 10 with this mutator on.
I picked Act 1 Chapter 1 and I got it in 5 mins.
Make sure you have custom skins equipped beforehand.