I'm looking for one or more biological processes that could be bolstered, changed, or introduced to a human to create a severely extended lifespan, ideally requiring death first, in order to best mimic the category of 'undead' creatures one sees in popular fantasy. Ideally, this process would also mangle the undead person's appearance and produce the classic undead drives of blood/flesh/brains, but these are a want rather than a must-have. The most important quality is that the process be relatively believable as a near-future process given what we know about science today.
The telomere is basically a protective layer for genes that prevents them from being truncated during cell division. After some time the telomeres are truncated so much that the genes will get truncated next - which means that the body dies. Theoretically by lengthening the telomeres you could achieve immortality - or at least very, very long life.
If you simply add that in your setting the not-yet found mechanism to lengthen telomeres requires certain... adaptations... to the human body you could have your non-evil undead. Perhaps each time the telomeres are lengthened your human loses some parts of what makes them human.
You used the vampires tag, so possibly the therapy would include drinking human blood to use something in the blood as a catalyst or building block for the lengthening process. Over time the people are getting used to it. They also get special medicine so that they can safely drink as much human blood as they need - or want.
In the end it's still a lengthening - not necessarily true immortality.
The very, very long lifespan might also have an impact on their brain. You probably don't know everything that happened when you were five or six - and just like you forgot about those years, your human would slowly start to forget about the ages twenty to one-hundred-thirty. The ones who focus on blending into society would be able to adapt - but some of them would become weird because they try to stay in the past or because life gets so fast that they can't keep up, just like a year was endless as a little child and is far too fast as an adult.
In the end, some of them might seem braindead - if they don't take care. But that's their fault - nothing evil to see, just humans neglecting their own mental health.
The fungus Ophiocordyceps controls ants so they climb down, when they reach the final position, the fungus forces them to stop and the fungi invades the ant completely.
Toxoplasma gondii is a parasite invading mice so that they lose the fear of cats and, gulp, the cat is happy. Unfortunately it seems also to invade the cat and it is suspected that they also can invade the human host, causing the cat lady syndrome.
Xenos vesparum attacks wasps and forces them to retract from their social duties and fly to specific destinations.
So, your brain parasite infiltrates the human brain. It reproduces by invading the human, so that bites and scratches can infect other humans. Because the brain is not longer needed, the parasite can also shut off the warm-bloodedness and slow down the metabolim extremely, leading to an extended energy reservoir (in fact most of our energy is needed to hold 37 °C temperature), very limited breathing, no detectable infrared radiation and undetectable pulse. They are "undead", but very much "alive" if a chance to infect another human is given. The crave for flesh is simply a byproduct of the infection. If no food can be gathered, the parasite extend lifetime by searching a cold place and going into an very long hibernation, it does not care if the human flesh rots.
You could fulfill your first 2 criteria by having a human mind in an artificial body. You side step death and decay that way.
Robot bodies are not uncommon in SF. If you tend towards the fantasy you could have a body like a golem: clay or stone, possessed by the spirit / mind of a person whose body was dead.
1: Without flesh to decay, the mind is immortal (if you want it to be).
2: The body would not decay because it is not meat. It might need a touchup with new clay now and again.
3: Re craving flesh - maybe there are deficits inherent in a clay body that make the mind in it feel strange irresistible cravings. Instead of flesh (it has been done so much!) you could have the artificial creature crave salt, which seems more golemy somehow. Blood has a lot of salt...
Conservation of energy seems to me to be the big one. While you can make zombies which eat an acceptable diet, most of them are depicted as eating a little meat here and there, and most are too slow to catch their prey. If energy was no longer a fundamental limit, all sorts of strange things occur. Perhaps these magical zombies have a limit to power instead of energy -- they only get so many kJ/day from whatever magical source keeps them alive.
Consider the parable of Two Wolves, typically attributed to the Cherokee.
A grandfather is talking with his grandson and he says there are two wolves inside of us which are always at war with each other.
One of them is a good wolf which represents things like kindness, bravery and love. The other is a bad wolf, which represents things like greed, hatred and fear.
The grandson stops and thinks about it for a second then he looks up at his grandfather and says, “Grandfather, which one wins?”
The grandfather quietly replies, the one you feed
Imagine if you lived to this parable all your life, not feeding the bad wolf. The bad wolf is constantly limited by starvation, and yet it persists. As the parable mentions, the two wolves are constantly at war, and merely not feeding one doesn't change that. Now imagine one day that starvation limitation was lifted. It could be a conscious choice, or a spontaneous discovery of a new energy source, or even magic. Maybe the bad wolf just found a way to burrow into the good wolf's food stash. It doesn't really matter. Now you have a wolf that, all it's life has learned to contend with the good wolf while starving, and it suddenly is not starving anymore.
That leads all sorts of places now, doesn't it?
To my eyes, you are describing a vampire.
- immortality: check
- slow or no decay: check
- classic undead drives: check
Now notice that vampirism isn't necessarily a choice. In most literatures, it is forced upon a person, usually in very traumatic ways. You have an intelligent creature that drinks blood in order to survive - doing evil things is entirely optional.
One might say that they should have at least some basic human rights. So they need some blood here and there... But hey, blood donation is a thing. Or maybe they can get it from animals. For a well-meaning vampire, it is only when access to these is negated or non-existing that they will turn to violence against humans.
A lot of twists can be made out of these. They might be an accident from science, that secretly fight to protect the innocent. Or they might openly be humanity's last hope against something worse. In either case some moral ambiguity will exist - keeping them from feeding will cause the death of innocents.
As to the changes necessary to make someone a vampire: take Secespitus answer, add furhter genetic alterations to include regeneration abilities (take some genes from salamanders or some other animal), and optionally add genes for porphyria, the "vampire" disease as well. The genes might be inserted into one's own genome by using some sort of adenovirus that is transmissible through bodily fluids (i.e.: injecting a vampire's saliva into a victim's blood).
Heavy exsanguination of the victim is necessary in order for its immunological system not to kill the virus, and once the virus has done its job the regeneration kicks in and "ressurects" the victim as a new vampire.
I don't think you can have a 'one-size fits all' answer to traditional undead. To start with there are a number of very different creatures you could call undead; liches, vampires, zombies and ghosts would appear to be the main categories.
A Lich is basically just a human who cannot die. There might be some kind of madness involved but they are generally just seen as humans who have found a method to cheat death (though not necessarily ageing).
First obvious answer has already been covered by Willk - robot bodies with human minds transferred into them. The second answer is human bodies with minds transferred into them. Either clones of the original person or other bodies depending on the desired result. This provides a still living being that would age and decay, and possibly die if the lich doesn't upload their mind in time and place it in a new body. Or you could say the lich keeps a backup of their minds somewhere so even if their current body dies it is not the end of their existence. This also ties in with the idea of soul jars where a piece of the soul is kept to preserve the lich's life.
Vampires... are more complicated. Genetic experiment gone wrong seems the most likely. Increased speed, strength and regenerative capabilities but needs something in blood to survive or fuel those abilities. Similarly you could say they are robotic or cyborgs, an early model that required blood to fuel themselves.
Zombies tend to be mindless, so some kind of bacterial or fungal infection that reanimates a dead body and gives it locomotion and drive without in anyway restoring the mind.
If you wanted to stick with the theme of the liches you could say they are failed attempts, people whose minds became corrupted in the transfer and whose clone bodies failed and began to decay while still 'alive'. Maybe they are what liches become after they have transferred their minds too many times and gathered too many small errors in the process.
Ghosts are difficult too. If we stick with the theme of immortality through mind uploading then ghosts could be minds stuck in computers. Perhaps they have holographic bodies or maybe they can just control a futuristic set of smart household devices (turning lights on and off, operating kitchen appliances etc. typically haunting behaviour).
If you want to have a variety of undead and a central theme that ties them together than I believe you start with the drive to achieve immortality through the transfer of consciousness and work from there. Whether you want robotic bodies or grown clone humans depends on the kind of setting you want.
A highly specific cancer.
Cancer has the potential to grant everlasting life if we can control it. If a cancer somehow manages to renew all parts of the body, including the brain, you have the everlasting life part. If this regeneration process is imperfect on parts of the brain you can lose functionality such as knowing what food is good for you. The body already locks down and isolates cancers of the body in "birth"marks. Magnify this to the body isolating and rejecting finger-sized malignant growths that would otherwise wreck the body across the entire body, and that can temporarily cause parts of the bodies surface to starve and die off, only to regenerate haphazardly after the growth has literally been expelled of the body (which also would have a gross fase where it starts dying off). Now you have a continuously scarred, regenerating and on patches bleeding/dying skin fitting for an undead. Bonus is that even a neutral undead would still be confronted as a leper.
Ofcourse,.this cancer is 99.99% magic at that point, but this is the best sciency handwavium I could come up with.
It may be possible that life is really old. The temperature of background radiation was in the Goldilocks region only a few hundred million years after the Big Bang. And there is no guarantee that nothing exists beyond the cosmological event horizon - meaning there may be things from outside the blast radius of the Big Bang that work their way in.
With most of the 4.5 billion year life of the observed universe, or with the uncountable amount of time that has passed outside the cosmological horizon, it is possible that some very strange things may have evolved out there and come here trapped in meteoric glass or as dust.
Human kind is an interesting species structurally. It begins as three sheets of extremely different specialized cells: nervous, digestive, and skeletal/circulatory. It's almost as if the human species is three different species in a symbiotic relationship sharing one common genetic library.
But it doesn't end there : some of the most fundamental building blocks to all cellular life don't come from the human genome at all. In this strange world - mitochondria, an ancestor of ancient bacteria, provide power and proteins for most animal cells. The process by which ancient mitochondria may have invaded animal cells and settled there is called endosymbiosis.
It is possible that a similar bacteria-sized endosymbiote, maybe even one specialized for human kind, is in the environment, trapped in a glacier, recently arrived, or so on. This bacteria, specialized as it is, might not be capable of infecting a healthy cell.
During apoptosis and autophagy, cells wind down their existences in an orderly fashion, taking apart the nucleus and inner workings of the cell before giving up. However, when animal cells die due to injury (necrosis) the cell struggles and eventually fails to hold it's inner workings against the forces of osmotic pressure. During this time the cell membrane begins to fall apart in pieces as it is compromised. This might even be how mitochondria originally entered animal cells. And our hypothetical specialist bacteria may be able to come in and set up shop.
Mitochondria quickly earn their keep providing cell energy and proteins. Endosymbiont X (for lack of a better term) may likewise help organize the dying shop. Maybe providing a great number of traits similar to Tardigrades, which include protection from starvation, dehydration, radiation, and shock. This might lead to the lungs being repurposed as a hemocoel. A hemocoel might require filling from some of the classic undead foods.
Endosymbiont X might also be able to produce factors that protect telomeres from drop off. Telomeres are the winding that protects DNA from unravelling, and is believed to correlate with cell life.
The new proteins could lead to structural changes in a human host, causing it to become deformed. Or, cell takeup of Endosymbiont X, as it depends on some pretty optimistic timing, might leave a good portion of the human actually dead, until those parts drop off or are removed.
The zombie uses live cells for self repair by first converting the consumed cells into a stem cell. This cell is then used for self repair.
Zombies prefer flesh and brains because those cells are easy to convert to the required replacement cell.
The answer by @Thorsten S describes how one is turned into a zombie. I'm only describing why they eat what they eat.
They are parasites that populate the outside of a creature's skeleton in my novel. Liches are basically the queen, a matured parasite, so liches can raise walking skeletons. They can also populate the inside of an armor, giving birth to walking armors. They form complex imitated muscle system to give the appearance of a walking person, however they are slower and more rigid. Ghosts are consciousness formed by time paradoxes, in another dimension, projected into our world. It's the answer to "what if I go back in time and kill my grandfather". Time and dimension get confused and they continued to project the person as living, but they are actually dead.