Good solutions. Mine is probably about the same amount of time, but doesn't require any special character configuration or change to the difficulty setting. It can't be done hands-off as with the drones method, but it's much safer than picking random close-encounter battles (trying for the achiement through normal play).
There's a mission not too far into the game where you have to rescue a Primarch from Paleven's moon. There's a place in this mission where you climb a ladder and take control of a mounted turret. At that point, two waves of Husks start approaching the barricade, and you pretty much just blast them away from a distance with the mounted gun.
If you have a save before this mission or somehwere near the beginning of it, find it and then make another save just before climbing up the ladder. That will give you a load point between rounds.
Load, climb the ladder, blast both waves of Husks, and then when the cutscene kicks in with the Brute knocking you off, just quickly skip through the cutscenes (takes about a second or two to skip) so that you see the "saving" message. Now just reload the save before the ladder, climb up, and repeat.
I seem to be getting 25 to 35 kills each round, and the round doesn't last long if you get into a good groove. I haven't timed it, but I seem to be getting about 500 kills in under half an hour using this method.
Maybe not the best, buf it might work for somebody close to the 5000 target and just needs a few hundred more.
(Update) Yeah, I timed it, and I got about 35 kills in less than 2 minutes. Factor in the loading time, and it should be possible to get somewhere around 500 kills every half hour. Gets boring after a while, of course.