This quest is triggered by entering Ulysses' Temple for the first time during the quest The Divide. Follow the short path to a terminal. It can open the door to the main hallway, which leads to Ulysses, and also to the maintenance bot room, which contains ED-E. Rescue ED-E for the optional part of this quest. Then follow the main hallway to an elevator down to Ulysses Temple. You now have to "Deal with Ulysses".
If you decide to fight him, be aware that Ulysses has a great amount of health, is capable of using Stealth Boys, as well as flash-bangs. Ulysses' eyebots are also capable of healing his injuries and providing fire support. One way to take care of the eyebots is to hack the eyebot terminals, one in each pit. For more details, see the article Ulysses' Temple.
If you decide to spare his life, you have to pass a hard Speech check of 100 (90 if you have a high reputation with the NCR or Legion or the Strip), which makes Ulysses join you against the marked men. If you have obtained all of ED-E's upgrades or all of Ulysses' holotapes, you do not need to pass the Speech check. Be aware, however, that if you wish to spare Ulysses, any dialogue that suggests deception on the part of the Courier, or attempting to use the holotapes option without having collected all six, will result in Ulysses being disgusted with the Courier. The only remaining option will then be to fight him to the death.
Marked men will show up after the battle begins, or after convincing Ulysses to fight beside you. Once the marked men are dealt with, you may either attempt to stop the warheads, or launch them yourself at a target of your choosing. This ends the quest and will trigger The End or The Apocalypse, depending on what you chose.