I want to run a short campaign that starts with the dead PCs' souls adventuring in the afterlife in Kelemvor's realm. Are there rules already for this? If not, what adjustments must I make to the rules for the PCs being dead when the campaign begins?
I have done this in 3.x extensively and, for reasons that will become apparent, I think the situation is similar. There are two funadamentally distinct ways to handle it that I have used, and both are appropriate for 5e play. Ultimately, the difference comes down to whether PCs are basically unchanged by death and the formation of outsiders (that is, the inhabitants of the outer planes who are formed from mortal spirits) from souls occurs very slowly over time or perhaps only by certain rituals or something, or whether that transformation is essentially immediate-- the dead quickly taking on the least form of the beings appropriate to their alignment and losing most of the abilities they had in life.
5e doesn't come down on which of these you should do. The DMG planar section is full of optional rules for outer planes that sometimes talk about planes being full of the 'spirits of the dead', like when discussing the recently deceased booking passage over the river Styx or the inhabitants of Carceri, but also talks about Hades being full of larvae formed from the souls of the dead. The fact these are all optional rules makes it seem to me like they intend you as DM to pick one or the other or a weird mix of both when using the default cosmology.
In any case, these are very different games. In my experience, I usually ran things the first way, where there's not much difference besides a 'dead' tag and a lack of body/soul duality in 3.5, while running things the second way, using the petitioner template, in Pathfinder. The mechanics of such campaigns will be very different from each other in 5e as well, since the differences between player and monster mechanics are very pronounced.
In the dead people don't change version, the game runs pretty much like normal, with a few caveats:
Being a single soul/body unit makes certain spells and effects function differently. Consider what you will have happen if a character is killed a second time before they return to life. Remember that things can't suck the character's spirit out of its body any more, or anything like that, instead either not working or leaving no body-missing-a-soul behind.
The character's corpses are still important even after death. If they are to be raised, you need to figure out what 'free and willing to return' means, and how the character experiences that. Is a character doing time in Bytopia for public drunkenness 'free'? Is a character currently struggling to escape from the pin of a three-headed guard dog of the underworld 'free'? What about a character whose alignment has been shifted by planar effects? What about a character who's failed a bunch of saves and been turned into a larva? what about a character trapped on Carceri? And then there's the matter of willing, and what sorts of information about the circumstances surrounding the casting the character is aware of, and if they get, like, a pop-up ad for resurrection or a mental query or the magic just figures it out inerrantly or what.
Characters of certain classes may have more dire problems. The warlock class, for example, provides several opportunities for a PC to create a character that is really, really dead once they are dead.
You'll also want to be ready to apply advantage and disadvantage liberally, in lieau of modifiers in 5th edition's rules paradigm, to account for stuff that's less or more of a problem when somebody's dead.
And, of course, be ready for the PCs, once they are high enough level, to go back to the material plane, pick up their bodies, put them on ice in some temple, and resume wandering round as normal adventurers who also happen to be dead. Of particular importantance is deciding what sort of thingy is required to know somebody is dead when you interact with them. Generally, I have people automatically know that, but that's a very 3.x thing and not very 5e-y. You might require a passive Wisdom(Religion) check, or something. In any case, remember that people might not know that before meeting the characters and the characters might very well not tell them, so then you'll need to know if they know or not.
The second method is much further away from normal. In Pathfinder, with the Petitioner template, PCs lose pretty much everything and instead have a very short list of alignment/form-based special abilities, and 2 levels of racial hit dice in Outsider. That puts them on par with first level characters, but weird. For example, they'll have many more skills than a typical first level character, and at higher levels, but vastly inferior combat abilities to any respectably built first level character, as well as no real option to be a spellcaster.
If you are interested in this method, the way to do it in 5e is to have players play monsters instead of player races. Design petitioners for each of the 9 alignments (pay special attention to True Neutral, which will need its own whole deal, cause there's no True Neutral Outer Plane, unless you count the Material Plane itself), give them a special thingy or two each (perhaps using these as inspiration), and have players pick what alignment they had in life. Just as your 12 skill points and one feat helped add in a tiny bit of agency in the 3.X version, consider letting players still pick a background and get the benefits and proficiencies that are a part of that.
There's yet more problems with this method, though, because you then have to decide how the PCs advance. You can either let them add class levels, which I would discourage, or let them evolve into more powerful beings of their alignment, which is okay except for balance issues. In practice, I've only used this method for specific one-shots or short campaigns that focus around ensuring the players will work together for the duration of the arc despite their different alignments being in the forefront all the time.
As for printed 5e material, you'll definitely want to check out the Planes section of the DMG, as well as what few outsider-y monsters have been published so far, pretty much all of which are in the Monster Manual.
I've not seen anything to address this whole subject officially. The answer from Dark Wanderer is excellent as all of those things are mistakes that you'd learn from after making them. There is no good answer specific to RAW, as 5E is too young to have produced a book addressing the complexity of the Planes of Existence.
If you dig into the Planar material, you'll get bits and pieces that relate to what an afterlife has to handle. Earlier editions have books on this, but know that each has a slightly different cosmology and vary from one another in meaningful ways. For instance, editions have flipped functions of the Ethereal and Astral Plane and which one is a path to the outer planes. I"ve tried to address this myself and I find there isn't enough meat in 5E. I find it convoluted as the pieces feel as if they weren't designed with the idea of all working together. Going back to other editions for source material--leads to some more confusion.
Since 1st edition AD&D, the outer planes match a spectrum of alignments--some of which did not exist in the game. There are 8 non-neutral alignments, but there are 16 outer planes, with half-steps between each alignment--half steps that do not exist in the game. In 5e, they overlap two alignments, like a DMZ between alignments. They mostly re-used names of various early European "heavens" and "underworlds" (valhalla, olympus, etc). By 5e they generalized them somewhat (Arborea, the Olympian Glades of Arborea, CG; DMG pg 58) But if you don't use these pantheons... the names become more meaningless. And are all Greek gods ChaoticGood? Olympus was the home of all the Greek gods. It is a confusing subject. SO don't hesitate to move away to something that makes better sense. It is the afterlife, it doesn't have to make sense to the living.
5E DMG pg 43 starts the discussion about the Planes. In addition to some flavor and mentions of places from past editions, they provide some possible effects on living characters that travel to the planes. The book is quick to say make this into your own.
But you're concerned with the dead not the living traveling there... the 5e books give you some referential ideas if you read through stuff.
In 5e The Prime Material Plane has three overlapping planes of existence. Think of it like phases in Star Trek. You change your phase of existence and shift into the Shadowfell or Feywild or Border Ethereal. Your body moves into that plane, but you are still where you were in the Prime. When you go ethereal, you see the material world you left around you. However, your body is converted to etherealness... incorporeal-just like ghosts. If you phase into the ShadowFell or FeyWild there are corresponding landmarks that relate to the same place in the Material world (and your physical body comes with you). FeyWild has evil and good creatures and regions. Shadowfell is only undead and evil. They are supposed to be echoes of the real world--but the design was more flavor than symmetry. For instance, the Shadowfell has gates to evil outer planes, but Feywild doesn't mention any access to the outer planes.
I am certain that I read that souls travel to the Shadowfell when they die. I can't find that in the DMG. This was true in 4E, maybe I'm imposing old editions on new. In 5E the portals to the shadowfell overlap where spirits and death lingers, such as battlefields, graveyards. So this sounds like there is a natural transition tendency to the ShadowFell at death.
Now the confusion for me in 5e, is that the Ethereal is not where your soul goes/exists, despite ghosts and specters being ethereal beings. In 5e, the Astral Plane is an infinite plane with no size. DMG.46, "visitors travel as disembodied souls to reach the outer planes." Pretty much sounds like this is where you go when you die. The Astral Plane is the realm of thought and dream (DMG.46) The Ethereal does not get you to the outer planes as it did in previous editions. Astral projection, PHB 215, sends your "soul" (astral body, consciousness) to the Astral Plane but leaves your body behind with a silver cord linking your soul to your body. If your astral form dies... 0 HP , you go back to your body via the cord. Cut the cord, your "soul and body are separated, killing you instantly." The vagueness here is "killing you," which you? Astral Soul "you" as well as your physical body? Death of the body is definite. But whether this kills your astral body/soul is the question for me. In previous versions, it killed the body and your Astral form was left to wander.
The weird part--which has always been confusing--is if you travel to another plane via a portal in the Astral Plane, your body is transported through the cord to the Outer Plane you travel to. This is how Gate works, you step through with your body. If you gate to the Astral Plane, your BODY moves to the Astral Plane as well--no cord. These two different forms of existence within the Astral has existed since early editions. The flavor of astral projection conflicted with the concept of just stepping into it. Also the Gith physically live in the Astral Plane, where you can find corpses of gods.
Maybe the Ethereal might be thought of as where lost souls wander the prime material? It does have portals in the Deep Ethereal to the other INNER planes. The books skip what the function of the Ethereal is in this edition. It has to for some monsters.
MM has bits and pieces in demons and devils, particularly about Lemures and contracts of owning a soul. Night Hags are a little confusing as they ethereally straddle a sleeping victim and invades their Dreams with a Nightmare Haunting Action. (Thoughts and dreams are Astral subject matters). So here the Ethereal plane is just there to ghost-walk through the world. The haunting can kill the victim. The hag tries to corrupt their thoughts and get them to commit evil acts when awake. If they do and the subsequent hauntings kill the victim, she traps their soul for transport to Hades.
It really is too big a subject for the core books. But the above are the official tidbits that I recall touch on the subject.
You really will have to homebrew it or borrow modified ideas from previous versions. Personally, I like the previous versions where the Ethereal was a ghost realm, overlapping all the Inner Planes and the Border Ethereal was the edge where you transitioned to the Outer Planes. The Astral Plane was solely a plane of mind and thought, never referring to the soul. Then the Gith started physically living there, god corpses... it got confusing too.