This achievement is a lot easier than one might think, especially if you have a friend you're playing with. The reason I say this is because all damage taken during or before the first helicopter goes down does not go against getting the achievement.
Start a survival mission on Resistance.
Before the round 6 starts, grab a predator missile (2,500 credits) and make sure you have a few hundred rounds handy. The two-story building by the air support vendor provides good cover -- head in there. After the round starts, concentrate on shooting down the chopper with your primary. Damage taken during this time does not disqualify you from the achievement. After the first helicopter is down, avoid getting shot by the second and nest yourself within the kitchen of the building. Launch the missile and take it down. Wham, bam, thank you ma'am.
If you have a friend, he (or she, to be fair) can run around and draw the chopper's fire if you find yourself struggling with it.