I'm currently still working on this and on mission 5. There's no "specific way" to get this achievement, but I will give you some tips.
** = Strongly Recommended
* = Recommended
**If your getting this achievement in Co-op like me, then you should use this strategy for EVERY LEVEL!
It doesn't matter if your doing this with 2, 3, or 4 people.
Have 1 person stay back at a safe spot while the other person(s) go up and kill enemies, the first person to die stays back and the other person goes. This helps very much!
**When fighting lots of enemy's in a small area try to only kill a couple and find a route to run through and trigger a checkpoint. The only time you can't do this is if it says you have to "Defend" something. (When you "defend" stuff you just have to kill all the people)
*If you find a vehicle and are headed to an area filled with lots of enemy's then save the vehicle until you've killed about half of them then use the vehicle so you don't go out and get killed in one shot by a Wraith.
*Try not to use "good" weapons right away (Sniper, Rocket Launcher, Shotgun, etc.) This helps because if you save it and don't run in and end up using it on some Grunts your gonna regret not saving it for an Elite/Hunter.
Those are some basic strategy's that I use and it's making Campaign go alot faster for Local Co-op. But these still work on Xbox Live Co-op! Let me know if you have any more tips I could add.