Mechanically, the main features of undeath are poison immunity, darkvision, not needing to eat/drink/breathe, and the undead creature type. Nothing there is particularly overpowered. In 3E, there was a "necropolitan" race which pretty much just had those things. You could whip up a homebrew undead race, give it those features and a few other perks for flavor, and come up with a ritual the PC could use to become a member of that race.
Note, however, that wizards* already have a way to achieve immortality--the clone spell. You can set the clone's age as young as you like, then kill your aging body, and your soul will transfer to the young healthy one. By normal human standards, this is far superior to undeath; not only do you stay young, you can still enjoy the pleasures of being alive! If what the player wants is a PC who is obsessed with cheating death, you might point him to clone.
However, if he wants his PC to be chasing undeath instead, it's worth asking why the character would choose the darker path. I can think of three possibilities:
1. The PC is already old and doesn't think he'll make it to 15th level before dying. Obviously, this rules out lichdom, which requires reaching an even higher level. He needs options and he needs them fast. The ritual to become a "bare-bones" (heh) undead seems like a good solution here.
2. The PC doesn't just want immortality, he also wants the power that comes with becoming a lich or vampire. For this, I think the player needs to be resigned to having his character become an NPC if he ever achieves his ambition. By definition, that kind of power would grossly unbalance the game.
3. The PC is insane and believes that undeath is a preferable state to life. This again would point toward a ritual to become a member of a no-frills undead race.
*Since lichdom was being considered, I assume the character is a wizard. If not, of course, clone may not be available.