To get this achievement, you simply have to keep using a certain type of weapon (pistol, smg, sniper, etc...). The more kills you get with a certain type, you will notice on your character page that it is leveling up and all weapons of that type will become more powerful. In order to get to level 10 you need to stick with one type of weapon for a long time, obviously whichever one you feel most comfortable with. I suggest going for level 10 with pistols, because both repeating pistols and revolvers count. This is especially helpful because you have two ammo types to use, instead of having to switch to some other weapon class because you run out of pistol ammo.
Edit: Now that I've had a little more time to play the game, I've noticed that you don't actually need to kill anything for your weapon proficiency to go up. If you have a friend that is of significantly higher level than you, hop into their game and follow them around with your favorite weapon equipped. By them killing the high-level enemies, your weapon proficiency will level up rather quickly and you don't even have to work for it!