This achievement is cumulative, so if you die the kills you previously got still count towards the total. You start the mission off with the XM25 (High Tech Grenade Launcher) as your secondary weapon, so press right on the D-pad at the start as you'll be using this for most of the mission.
Make sure that you keep it on you the whole time, since you won't find another one. Now just make your way through the mission only using this gun to kill the enemies. If you run out of ammo, swap back to your starting weapon, but don't worry as later one you'll come across an ammo crate and auto refill. If you don't have it by the end, simply reload the last checkpoint before you complete the mission and get the remaining kills.