The Black Eye of Obsidian achievement is unlocked after defeating Sky Slasher, the Obsidian Cult overlord. Defeating this boss concludes the story of the Definitive Edition Title update for ReCore. It is story related and cannot be missed. Post-game play is permitted after defeating this boss.
Story related activities required before this encounter include:
--Unlocking the T8-NK frame in the Renegade Vault responding to Violet's tasking
--Completing the Tumbledown Dungeon in The Cradle to gain access to the Starving Sea
--In the Starving Sea, completing the Brimstone Cut, Well of Shadows, & Scalding Cell dungeons to gain access to the Obsidian Eye dungeon where the Obsidian Cult overlord resides
Defeat the high level enemies outside the Eye of Obsidian Dungeon. There is a fast travel point just south of the dungeon that provides the most convenient access. Eye of Obsidian is a level 36 dungeon but it requires only 12 prismatic cores to access. By virtue of making it this far, you will have well exceeded this number. Once you enter, there is a fast travel point just inside. Use it to adjust your crew to your liking. From your entry point, in a corner to the immediate left, is a health booster. This is the only collectible in the dungeon. Enter the main hall.
After a cut scene, Sky Slasher appears. He will be assisted by some other corebots as well as by flying healers. I recommend Mack in an SP-DR frame. That combination makes available a sweeping laser attack that inflicts substantial damage to all enemies in the arena and does not have to be aimed. The fight is not difficult providing Joule evades Sky Slasher's devastating attacks.
Defeating the boss, triggers a cut scene during which the achievement pops. Below is a video showing entry to the Eye of Obsidian dungeon, the dungeon's fast travel point, the health booster, and the boss battle. Following the achievement popping, the CoreBot loadout is shown.