Once you get to mission 5, pretty much just follow the waypoints. You can go stealthy or a mad dash. Either way, make your way to the Eye of the Dead God (getting Dead Eye in the process). Once the cutscene ends, turn around and you'll be in the void. Go over to the telescope on the cliff and interact with it to see a new waypoint leading to the Outsider.
If you're careful, you can make your way down the cliff using a combination of falling and Displacement/Blink (if playing Original Game +). Once at the bottom, go through the portal (at the waypoint), and then follow the last waypoint to the Outsider. You'll have two options. Pick the first, and kill the Outsider and get the achievement.