If you want to get this achievement as fast as possible with a new character, just do the initial sidequests and story quests until you hit level 5. Around the same time you hit level 5 (and get your Turret ability), Sir Hammerlock will ask you to go to Southern Shelf Bay to kill Midge-Mong.
As soon as you enter this area you'll see some Rakk flying in the distance. Fire a shot to attract them, put out your turret, and stand back and let your turret handle them (since if you kill anything you'll "steal" the kill from the turret). Rakk are extremely weak, and often fly in a straight line towards your turret making them easy targets. This first pack consists of around 6-9 Rakk in total.
Once they're all dead, just Save/Quit to main menu, and then hit continue. You'll enter the game just by the Rakk again, ready to repeat the process. Repeat until achievement pops, which took me about 20 minutes after I got the ability.