Lost Souls are NPCs that can be found in any area. The Marshall (your character) will make a comment when one is near. Interacting with them will allow you to accept their Burden. Doing so drops Iron and Sin, and possibly gear. You will get a skull over your head, with the number 6 underneath, meaning you are Burdened until defeating 6 enemies. While Burdened, you take triple damage. Accepting multiple Burdens (e.g. finding another Lost Soul before clearing a previous Burden) will add 6 more to the kill requirement.
You only need to accept the Burden for it to count towards the achievement, it's okay if you die before getting 6 kills.
The 3 Righteous Souls that are required for the "true ending" do NOT count towards this achievement, only the randomly spawning Lost Souls count.