Build your relationships with Nyx, Eurycide, and Orpheus when possible (with nectar and conversations). At some point after you gain access to the administrative chamber, a Nyx dialogue will prompt the ability to alter Orpheus' contract so he can be free.
Go to the admin chamber and grab the contract on the back wall. Bring it to the House Contractor and he'll alter it for the cost of 3 diamonds.
Play through another run. When you return, Orpheus will say something amazing happened to him and he'll leave. The prophecy should be marked as complete now. If not, you may need to actually meet him in eurycide's room in asphodel.
Go to your desk and fulfill the prophecy. The achievement should pop when you leave your room.