The easiest way to do this is to remember two things: kill the helicopters and get as many group kills/headshots as possible.
I went through the mission with a SCAR-H, 338 Recon, and the grenade launcher they give you to start. As you parachute in, pepper the deck with the launcher. Don't worry about ammo since you land on an ammo crate. Grab a sniper rifle as your off hand, ammo up, and climb the ladder.
Once on the deck, use the launcher and grenades to get multi-kills and the sniper rifle to get your headshots. Clear the enemies at the beginning and wait for the heli to come in with reinforcements. As soon as it hovers, spam the grenade launcher at it. If you cleared the area before, you should be able to get close enough to hit it no problem. That should take care of about 1500 points right there if done right.
Next, climb the ladder directly in front of you. It gives you an easier time to lob grenades into bigger groups from the higher vantage point.
After that, take out the second helicopter with the Stinger or grenade launcher and you should be really close to the 6000 point mark.
I hit mine while still on the deck, but there are still several baddies to kill below if you need a little extra. Good luck!