I've run a number of games where PCs became undead and/or immortal. The thing is, mortality is such a huge and canonical drive and characteristic of sentient experience that I sort of think it's...bad roleplaying... not to include some downside to losing it. Powerwise, it's the equivalent of maybe a magic item or two. Some thoughts about it:
* It's easy enough to create your OWN type of undead, rather than looking toward WotC to supply all the answers. Some ideas to throw around: Shriveled but preserved undead, maybe something like a mummy. Sentient zombie or skeleton-like being. Twisted, ghoul-like creature or something else that has become infected with some sort of magical disease. Non-undead ideas might be something like a sentient golem or a creature that has been transmuted into some sort of angelic being. Perhaps a god makes the PC immortal after the character performs some cosmically important service. Or agrees to everlasting servitude. Maybe even something incorporeal...but this one really needs some balancing downside (like a limited ability to affect the physical world). I think curses and blood oaths are also thematic ways to become undead.
* What sustains eternal life? Maybe the creature needs to feed in some unnatural way - like consuming flesh, blood, the hearts of the living, magic of some type. Maybe it derives its life force from the faith that endows a deific being....as long as it acts in accordance with that being's cosmic goals. Or is gifted extensions of supernatural life when it performs services for a master.
* What was the experience of becoming immortal like? How does the experience of being immortal differ from being mortal? I've tended to imagine transitioning to undeath as being traumatic in some way, and have tended to hand out compulsions, flaws, types of insanity...or offered the player a choice of such. Doesn't NEED to have any sort of mechanical penalty, but it might. Examples: Character can't feel emotion and/or/including PLEASURE any more...does not benefit from any effect that might inspire it; and suffers penalties/disadvantage on diplomacy checks that do not rely on cold, hard logic. Character feels some sort of involuntary drive that is hard to resist - to kill, feed, patrol like some ancient sentry, or something else. Character feels phantom aches an pains from when they were living; or can't get used to not breathing and has panic attacks where they feel short of breath. Character derives no joy from the world and suffers constant depression. Character suffers violent or psychotic episodes occasionally - and is prone to attacking the living when distracted. Or sometimes can't tell friend from foe for brief periods of time. Character feels like dead bodies are still alive, and possibly sometimes trying to talk to them. Character can't sleep any more whether they want to or not or suffers terrible nightmares when they do; derives reduced benefits from taking a long rest. Character (if they've gone the celestial service route) finds themselves stalking innocents, just to make sure they're safe. Or finds themselves giving away wealth and belongings to those in need if not concentrating on doing otherwise. Character becomes completely and dangerously obsessed with some course of action (kill all the West Meritorians), item ("protect the sacred MacGuffin" or "if anyone defiles this magic nail piercing my shoulder, I'll die!"), place, or person (defend my king or lost love, regardless of whether he's still alive); probably relating to how they become undead.
* Depending on the type of undeath or immortality, some sort of physical flaw, vulnerability, or limitation might also be appropriate. I've seen examples like rotting flesh brought up above. Others examples: Hunted by from time to time by holy priests / druids / fey / paladins because they've become unnatural abominations. If anything dislodges or destroys the magic nail pinning their soul into their body, they die. Magically compelled to serve a master; or may have immortality revoked if they disobey a master. Has some sort of weird rule relating to their supernatural nature like can't step on hallowed ground, or can't cross running water. Maybe they're vulnerable to worms, maggot, insect swarms. Maybe they transmit some sort of disease whether they want to or not. Maybe they look hideous without some sort of disguise. Maybe they sink in water because they no longer of buoyant flesh. Or start to fade away if they don't work toward some goal or commandment every few days. Or devolve into animalistic non-sentience if they don't feed on flesh, blood, souls, etc every so often. Maybe they feel physical pain and distress (possibly even taking damage) when something important - like innocent children, the land they're sworn to protect, or their personal treasure horde is threatened or harmed.
