My take on getting Deliver This and the other three achievements for this mission.
At the start, pick up the Barret .50cal sniper and self-revive. Move up the the forklift and kill the guys in the building and then the two RPG guys in glass building slightly to the right of the other guys.
Move up to the far left wall, just behind the fence. Throw your first care package here and clear out this area. Watch for guys on the catwalk up on the top left and guys that repel down from the building and fast rope from the helicopter.
Once you round the next corner of the building to the right, watch out for 5-6 guys that rush you. They can be taken out with your M4 from behind the barrels for cover.
Move up to the left and the pillar that is closest to the way point, throw down a sentry turret that you may have got from a care package. Eliminate the guys down that walkway and wait for the first helicopter to come and drop troops on the other side. Remember to keep trying to get the XM-25 care package if you have not already gotten it.
When the helicopter starts dropping troops, shoot them out of the back of the helicopter before they fast rope down. It will cause them to fall to their death.
Once the helicopter troops are gone, move forward and take one of the packages that you need. Once you get it, prepare for a slight ambush and make your way back to the pillar and sentry turret. You will repeat this until you have all three packages.
Once you have all three packages and the area is clear, you can move to the extraction zone. On the other side of the room at the door, a group of soldiers and one juggernaut will show up, use you flash bangs and 50 cal to eliminate them. Then move to the lone survivor and revive him.
All this can be taken in from my video below.