For this achievement you must have a certain weapon type (SMG, Sniper ect) equipped when you obtain XP. You don’t have to actually kill enemies with the weapon for the proficiency to go up, just as long as you’re holding it.
The weapon proficiency progress can be viewed in the menu.
If you do have a friend who is a higher level than you, jump into their game with your weapon class equipped and have them kill higher level enemies, the weapon proficiency will quickly climb without needing much work from yourself.