This is a short guide depicting types of immortalities in Elder Kings[www.elderkings.com], overhaul mod based on The Elder Scrolls universe.

It contains both those meagre scraps of information I could find on the subject myself, as well as everything relevant that I had to discover on my own in game.
On a slightly relevant sidenote, "the answer is in our Discord"-comment posted years ago, should not be a valid answer to any post.
This is based on the latest Crusader Kings 2, all its DLCs and latest Elder Kings before they migrated onto CK3.

Whenever a character gets too old in Elder Kings, he has to get a savethrow every year or die from an old age. Imagine rolling 1d10 every year and, unless you throw nat10, you die. Even if long lived mers have a bar of around 300 years compared to human races with their meagre ~80 years, it still makes no sense for anyone powerful, skillful and ambitious enough to not pursue some form of immortality, unless it's for religious reasons like feasting in Sovngarde or prowling Hircine hunting grounds.

And indeed, it is even lore friendly in The Elder Scrolls, seeing such examples as ancient Dragon Priests, Tiber Septim becoming an Ninth Divine known as Talos, Mannimarco ascending to Necromancer Moon or the whole ALMSIVI theocracy.

Undead can be either Known or Unknown to the world. Undead are generally considered an abomination and have very short "life" expectancy if they become Known in any realm that adheres to religion that does not tolerate undead.

There are three different levels of tolerancy:
Undeath is considered sacred - small opinion maluses for Known undead (and actually small bonus for tame Benevolent vampires).
Undeath is tolerated - medium opinion maluses.
Undeath is seen as abomination - ruinous opinion maluses (something like -150 for liches, -100 for vampires).

Unknown undead are considered hidden and cannot be perceived as undead and do not receive opinion modifiers for being undead. Your undead courtiers have no visible to you traits of undead as long as they stay Unknown. Court wizard of modern Solitude, anyone?

Once they are exposed as undead, they permanently become Known.
Courtiers exposed in a realm that does not tolerate undeath can be executed at the moment of their discovery.

Note that Meridia smites her followers to ashes herself if they dare to become undead, whether they're Known of not.

There are four forms of undead: lich, vampire, draugr and skeleton.

There are two types of undead: independent and thrall.

Thrall is basically a mindless servant. It could be either a skeleton courtier raised by a necromancer or draugr-type undead that obeys a fellow draugr ruler.

Obviously, your ruler can never be a thrall, so let's discuss types of independent undead below.

Most (in)famous form of immortality, unfortunately in terms of gameplay it is mostly a "poor man's choice" unless you pursue absolutism and want to delve so deep into tyranny that you'd have enough power to abolish the whole concept of it inside your realm.
Lich was a living necromancer who commited ritual suicide to keep residing in his body after its death (ironic, isn't it?). Liches are completely infertile. Lich eventually *will* convert into draugr culture.

Have a magical education and be proficient in it.
Educations: a Journeyman Mage or better, an Expert Battlemage or better, or a Master Nightblade or Master Priest.

You need traits such as Cruel, Tyrant, Ruthless, Ambitious, Reckless, Envious and Impaler and cannot have traits Kind and Just.

You also need to be either Cynical or adhere to religion that tolerates undeath. Worshipping Meridia blocks necromancy regardless of anything.

I did not disassemble game files to confirm, this is based on EK dev's message here:
https://forum.paradoxplaza.com/forum/threads/how-to-necromancy.871152/post-19653441

Get a random event to pursue Necromancy in your studies.
As far as I know, you get an event to start on Necromancy based on the requirements listed above. Being a friend with a known necromant makes no difference.

Then go through a string of events as time goes by that lets you advance your studies in Necromancy with different levels of success. Failure to learn can permanently lower your health or make you Stressed.
If you join Guild of Mages and rise to be an Arch-mage, you can lift the ban on Necromancy. This gives a long-lasting opinion malus from all members of Mages Guild worldwide who do not tolerate Necromancy, something like -25 opinion for 50 years. This also lets you use Mages Guild libraries and your fellow mages' help in your studies of Necromancy which greatly helps you in your lessons.

Once you're an accomplished Necromancer, you can perform a Ritual of Lichdom and become a lich. Some RNG to fail the ritual and die properly, but seems to be open for savescum.

Liches are always Known undead.

Liches go through stages of decomposition with MTTH[ck2.paradoxwikis.com] of 25 years.
(None to light, light to advanced, advanced to full - took me total 120 years in my playthrough).

Slight decomposition gives everyone around -10 opinion. Medium decomposition gives -20.
While you're decomposing, you're a faux undead - you still count as your old culture and your government stays the same.

Final level turns you into genuine undead and that's where the things change drastically.
Your culture changes to Draugr. Your old culture forgets your birthplace and now considers you a foreigner. You're given a separate Undead trait and your fellow draugrs now have +200 opinion that overlaps most of the things you can do to them. This is the only way to play as draugr besides starting off as a ruler in Frozen North.

Your government type changes to Undead Horde, your laws default back to uncentralised tribals and you're locked into tribal stage without option to advance to feudal-imperial systems, you cannot create kingdoms or empires (can keep old kingdom or empire titles), this includes creating cultural empires like Aldmeri Dominion as you stopped being an altmer (or whatever your previous race was) when you changed to draugr.
Genuine undead is basically closed off club where all draugrs and skeletons get +200 opinion modifier, they do not oppose being imprisoned, nor do they object about being revoked titles, they pay full taxes, they basically serve you wholeheartly with no objections and in any practical way they're better than keeping your old living friends around.

Your threat level never decays, so nothing stops you from constantly launching wars except losing prestige from breaking temporary truces and you can now pillage captured holdings for gold and tech points. Your armies deal huge morale damage and are significantly slower, which explains their slow movement on the world map and their lowered flanking ability in combat, and they have horrendous retreat, which doesn't really matter because of their huge morale defense (morale can't be broken if your troops cannot comprehend what morale is *taps bony head*). Did I mention that, as a necromancer, you can raise huge undead armies and there is jewellery that allows you to raise even more troops, as well as boost their capabilities?
Thus, the naming of Undead Horse is apt. Becoming a full lich is a drastic game-changer!

Although there is a mod that lets you control a decay so you don't degrade into Undead Horde:
https://www.nexusmods.com/crusaderkings2/mods/81

Guild access - none!

Vampires are undead that contracted a curse of vampirism. They need to regularly feed on living mortals and suffer lower fertility, but still can conceive children.
Looks like the fertility rate locks on the age when the character became a vampire and doesn't go into "barren" stage with aging. Untested with female vampires.

There are two types of Vampirism: regular and pureblood.
When you're not pureblood it means you caught vampirism from a sickness (sanguinare vampiris), when you are pureblooded it means you got vampirism by either Molag Bal himself or have been turned by a pureblooded vampire and the bloodline traces back to Molag Bal this way.

/untested/ You can become pureblooded vampire by worshiping Molag Bal and trading in a bunch of favor.
Lore on interaction with Molag Bal is not... pleasant.
https://en.uesp.net/wiki/Lore:Opusculus_Lamae_Bal_ta_Mezzamortie

1) If you get a vampire infestation in one of your provinces, there's an event where you can be attacked by a vampire and infected.

2) One of your courtiers/vassals can be a hidden vampire but you won't know it until he's exposed. If your realm's religion tolerates undeath, you can keep him around after the exposure and befriend him, then ask for a conversion into vampirism.
/untested/ It's theoretically possible that you can befriend a hidden vampire making him disclose his condition to you personally with an offer to be converted - how do you even test that while they're hidden because you do not see their possible vampire status?

3) Cheats:
If you want to manually trigger the infestation-personal-attack event through console commands, the event id is "vamp.4" (This will work even if you don't have a vampire infestation).
You're not guaranteed to be infected from this, it may take a few tries to get the Wounded that turns out into infection.
Command is "event <event ID>"

"add_trait vampire" should also instantly make you into a vampire but it's safer to go through event-infection to not miss anything.

You start off as Unknown (hidden) vampire but can come out as Known vampire at any time through personal decision or get discovered by a third party later.
You cannot un-known it after the knowledge gets out!

If your realm tolerates undeath and your courtiers and vassals are aware of your condition there is a decision to offer them that, however they have to be friendly (can not despise because you are a vampire and need to have at least 50 opinion). If you are not openly Known, there is a random event that allow you to inform single random friend or family member about the vampirism.
/untested/ Forums say argonians cannot be turned this way? But we have argonian vampire clans: "the Whet-Fang sodality of Black Marsh, who use magicka to keep captives catatonic and harvest from them the red nectar" -https://en.uesp.net/wiki/Lore:Manifesto_Cyrodiil_Vampyrum

There are four stages of hunger, you need to feed regularly or advance to next stage through pop up event after sufficient "dry" period.
Hunger makes you lose diplomacy, but get martial and learning.
Not feeding on maximum hunger can give you Stressed, Wounded or Death by starvation but you get a warning in pop up before that.
Feeding removes Stressed and resets you to Stage 1 of hunger.
/untested/ Pureblood vampires can go longer without feeding, suffer less from hunger and do not have lethal hunger - Death by starvation.

You can feed at any time through personal decision.

Feed on locals - success depends on Intrigue. If discovered, bumps you into Known vampire, enrages peasants that you attacked - provincial debuff "angry locals" + possible revolt uprising.

Feed on courtier - sucks a courtier dry which, if successful, kills him. Seems to picks low opinion courtiers first. Success chance seems to be based on personal combat rating (same as duel). Courtier can fight you off, exposing you and also possibly contract vampirism in the process.

Offer to confer vampirism - offers vampirism to your friend (good opinion, suitable religion), counts as feeding. Your friend must already know about your condition - either Known vampire or told them before in confidence. Recipient becomes Unknown vampire even to you until he comes out. Funny but it's more reliable to tell vampirism by checking Vampire Scion on their children instead.
They will have to feed on their own and can possibly get killed during it.

Unknown (hidden) vampires suffer no changes from opinions.

Known vampires with religions considering undeath sacred get 0 opinion change, tolerated -50, abomination -100.

Known vampires can denounce their condition as a curse and publicly repent, becoming Benevolent vampire, that lowers negative opinions, lowers revolt risk, lets you feed on volunteers without murder. Must have good traits, not have bad traits (e.g. Kind, not Cruel).

As far as I know, vampirism has not been fleshed out enough in EK1 to have bloodlines, this comes with separate mod
https://forum.paradoxplaza.com/forum/threads/submod-vampire-bloodlines.1084661/

There are no drastic game-changers, you keep your culture and government type as a vampire, but you need your local religion to be at least tolerant to undeath or face local revolts.

Guild access - no restrictions if Unknown. Only Dark Brotherhood if Known.

Honestly, the most bland and tasteless option. Immortal is the powerful character who halted his aging permanently with a magical ritual, sacrificing his ability to reproduce. There are no other changes besides becoming infertile.

You need to be Legendary mage, have Learning 30 and bunch of accumulated magicka.

Good magical education + years and years of passive studying in Mages Guild. Without good magical education during the upbringing, there is a good chance to die from an old age before becoming Legendary. Mer races seem to have their education time increased proportionally to their life expectancy.

Learning skill. Can be boosted by Fortify Intelligence, Learning focus, artefacts.

Accumulated magicka. Passively restores with time. Can be cheated with "event magicka.1".

It is implied that immortality ritual pulls success chance from character seed (assigned at birth) instead of RNG to prevent savescum.
Mod to change that:
https://steamproxy.net/sharedfiles/filedetails/?id=3051923491

Part of Hagraven Cults in the Reach. Untested and I'm not even sure it has been implemented in EK or only in EK2.
I saw nothing in game without playing as dedicated Reachman, but there are trait icons in game files.

Done in few stages during March 2024 after I had to revert a savegame for ~120 years because becoming a full Lich locked me out of most diplomacy options and not a single place on the internet warned me about that.

Paradox Interactive for creating and expanding Crusader Kings II.
Bethesda Softworks, including everyone who left earlier, tentatively including Todd "Sweet Lies" a.k.a. "It Just Works" Howard, for creating The Elder Scrolls.
Elder Kings team for creating and expanding Elder Kings.
People from Paradox Plaza and Reddit for sharing knowledge.

The ending of the words is ALMSIVI.