Split/Second

The best cars to use for this achievement are the Ryback Coyote (Class B) and the Cobretti Pursuit (Class A). Both are excellent at drifting (rating of 9).

You'll get the Coyote first, with the Pursuit being one of the last vehicles to unlock. The Pursuit is faster and has better acceleration than the Coyote, but the Coyote has more strength, making it a bit easier to control during shockwaves.

The key to ending this event quickly is to build up your Power Play meter to red, and then hit the helicopter with a level two Power Play (B button) when the prompt shows up. Each successful missile deflection with a maxed-out Power Play meter takes off four life bars from the chopper; you'll have to hit it four times to destroy it. Doing it with low-level power plays takes too long, since these only take off one life bar.

To build up lots of Power Play energy, drift every chance you get; as mentioned, the Coyote and the Pursuit are great at drifting and you'll fill the meter really fast when you combine that with avoiding waves of missiles. When the helicopter icon glows red, hit B as quickly as you can.

Good episodes to get this achievement on are Episode 6 and Episode 8. Episode 8 (the storm drain) in particular is good because large sections of the track are very wide open (making it easier to avoid missiles) and there are a few really long corners to drift around.

For what it's worth, I got this achievement on Episode 6's Air Revenge sequence. Using the Pursuit I beat the target time by nearly six seconds, and that's even after getting blown up once.

Easiest way to do this is to pick a car you're good at drifting with and try to miss all the rockets the chopper shoots. You'll get about one and a half bars full from evading and perfect evasion. Drift around all the corners and save up untill you have the lv 2 Power Play and fire back at him.

The level 1's take away only 1 bar while the level 2's take away 4 bars.

The first Air Revenge event is in Episode 6: Midway Pass.
It's like Air Strike, except you can also attack the helicopter.
Use a car which is fast for better missile avoidance chances or one with very good handling, whatever you're comfortable with. As PickAxPete said, using Level 1 power plays may take too long since they remove 1 life bar from the helicopter. However, they may help if you want to get rid of the last life bar of the helicopter quickly or in other similar situations. To take away huge chunks of the life bar, charge the power meter to red and use Level 2 power plays. Try to focus on flawlessly dodging missile waves, performing perfect waves. These immediately give you a whole power bar, filling up an additional third of your power meter. Look ahead to see where missiles will land and plan the optimal path. Drift to avoid missiles if you know you can or see a clear path between them suitable for drifting. Drifting, avoiding missiles and doing other things combined builds up your power meter fast. Once you obtain a red power bar, use it on the next wave of incoming missiles to deflect all of them back. Continue with these steps and quickly charge and use some missile deflection power plays. To win the event you need to destroy the helicopter within a time limit. I managed to do it the first time with around 10-20 seconds left.