This can be done in Forge, it does not need to be done in Campaign.
Simply load up a forge game with a second controller signed in to another account. That account will be set to be an elite, which can be changed in the customization menu. Using your main account, pick up a respawn point, press
and click 'Delete All of These' to get rid of every respawn point. Then, place one respawn point anywhere on the map and have the second account commit suicide, so they will respawn at the respawn point. Simply beat down the elite from behind, step back a space, kill him again after he respawns. Repeat until you get 1000 kills. This will also count towards this achievement: