I haven't attained this achievement myself but a good idea might be to play insta-gib, you use the rail gun to kill your enemies instantly and you can reload almost instantly.
You can kill someone for 100 damage instantly and the max kills is 25, so you can deal 2.5k damage every round (note that is only if you get the max kills).
It's probably a better solution then other modes because kills take longer and are not instant. Note: Insta-gib is an LTM (limited time mode) it does return but I don't how long you have to wait.
Some alternative modes are team swat and team shotty snipers, these modes can get you quick kills. Team swat is a mode where headshots are insta-kill and the score limit is 50, so if you get a bunch of kills by the end that can help you progress. In shotty snipers you can use the sniper to one tap the enemy and the shotgun deals a lot of damage up close. Don't worry these modes won't go away they are not Ltms.