IMPORTANT UPDATE 21/7/17: WITH THE QUESTIONABLE CHANGES TO MULTIPLAYER THAT LAUNCHED WITH THE 6.66 UPDATE 20TH JULY 2017, THE BELOW IS UPDATED.
THE GRENADE LAUNCHER IS NOW UNLOCKED AFTER LEVEL 22. SIMPLY EQUIP THIS AND GET 10 KILLS FOR THE ACHIEVEMENT.
For this, equip the new weapon grenade launcher, and put it as a secondary weapon in a couple of your favourite loadouts. I wouldn't recommend it as a primary weapon, as it is not very multidimensional and is good for ranged assaults more that CQB.
I got this in one match of Team Deathmatch. This is most effective when you wait for other fights to start, coming in and spamming some grenades into the group and hoping you get the kill. Picking enemies up from a distance or ones confined in small areas is a good method as well. However way you do it though, 10 kills should come in 2-3 matches for most; for me, it came in one.
(P.S. Thanks to No. 1 Richie who pointed out that Private matches does not contribute to achievements.)