TimeShift

Me and my boosting partner found MP1 incredibly unreliable for this achievement. Instead, we used MP14, Sky Hook. It is much more open and thus it's easier to find your partner.

Start a Capture the Flag player match, with insta-kills, instant spawns and enemy radar on, as well as respawn protection turned off. Whoever is going for the achievement, grab a fast-firing weapon (Karbine works fine) and head over to your opponent's side of the map. Kill him, then use the radar to quickly find him again - he will spawn close by, since you're playing CTF. Just keep following the radar and quickly kill your buddy 5 times within 15 seconds for the achievement.

Note: Multiplayer achievements in the main game (non-DLC) only pop at the end of the game.

Boosting partner needed / preferred.

This can be done with just you and another person.

Host a player match (deathmatch) on MP1. Set rules to one shot kill, and make respawns instant, and starting weapon the assault rifle, it may take a few tries so set the kills to 25.

Now there will be a little room with a red light in it, one person stands right outside/ inside that room and aim down the hallway in front of you. Now kill the person, he should respawn very close to where he died, all he has to do is run to that hallway again and you will continue to kill him 5 times, you only have 15 seconds so both of you need to be as spot on as possible.

Sometimes the person will respawn in the building right next to an open window, when this happens it is very hard to make the 15 seconds as it is the farthest away spot from the hallway. This is why it make take a few tries.

I did it with another person and got it in about 2 minutes.

Note: You can increase your run speed before starting the match making this solution a bit easier.