See Pistol Mastery for more info.
SMG's are a very unique weapon. Instead of waiting for your crosshair's to turn red like the pistol or the shotgun, you must increase your damage multiplier as high as you can, which is 1.7 damage. Simply hitting the target is all it takes to increase your multiplier, however it can be reset in one of two ways. If too much time passes between your last hit, or you reload, your damage multiplier will automatically be reset. Any upgrades you can get that will increase how much ammo you have per clip will make this achievement much easier to go for, so try to get it if you can. Also, using the skill Bullet Storm in the SMG tree will give you infinite ammo for a small period of time, which can make this achievement much easier.
It's hard at a range to try and pull this off unless you are invested into the SMG tree, so get as close as you possibly can to enemies, and slowly shoot quick bursts into them. Every time you hit them, you will notice a bar at the bottom of your crosshair's fill up, and your multiplier to the right of it. You may have to kill multiple enemies for it, but once you hit a 1.7 multiplier, you can just let it go.
You will unlock the perk "SMG Fury" after maxing out your damage multiplier four times.