RAW, this isn't possible.

There are limited choices available for RAW races, and being an undead skeleton isn't one of them.

Given that you are just starting out as a DM, I highly recommend sticking with the vanilla rules as much as possible as you learn the system. This does mean asking your players to limit themselves as well to what they can build normally. And while there may be playtest material (like Unearthed Arcana) out there that fits a desired theme, I think you might have an easier time either reskinning for flavor an existing race, like the Warforged Dan B's answer, or working with the player on a new concept.

But it doesn't mean they can't have goals

Your boyfriend can still want to become what he's asking here. And it sounds like he kind of wants to be a lich. Here is a good question about how a PC can become a lich.

That's definitely an aspiration (which is awesome, character goals make roleplaying interesting), but also be wary of your table dynamics. If you aren't running an evil campaign (which I also recommend against for your first time DMing), then this isn't a great goal for him.

Making sure you and your table (including your boyfriend) are having fun is the most important thing. But it's also your job to make sure someone's fun isn't ruining someone else's. This sometimes take place in the form of My Guy Syndrome. I'm not saying that's where this might be heading, but it's something to be aware of.

Wizard - Necromancer

Just a quick heads-up that if they do go the route of Necromancer, or any other class that uses minions, you'll need to account for those minions in both your world and your encounters. Be fair (but still account for your world) in both cases, but in the latter you will likely need to step up the difficulty.

I've got a friend playing a Necromancer and he doesn't take his minions into cities or towns. It makes it a bit harder to get them up and going in combat, but he tries to be realistic about how the world will react to his skeletons.

Begin with a Session Zero

A good way to get everyone on the same page is to run a Session Zero.

It's a little bit dangerous to experiment with custom rules as a beginning DM. The risk is that you hand out something that's too good, and then one of your characters is more powerful than the others and it creates tension.

(In particular, we have an existing antipattern where the DM invites their significant other to the table and then makes a bunch of rules exceptions to make their significant other's character super badass, which tends to annoy the other players. You'll want to be careful of this.)

The best way to generate a custom race is to "reskin" an existing race. In your case I'd recommend reskinning the warforged race (WGtE, p. 68). By default, this is a race of robot warriors who have somehow gained sentience and humanity. You can go through the rules and replace the word "robot" with "skeletal" and you'll get something reasonably close to a skeletal warrior. In particular, your race would have:

Warforged Resilience

You were created to have remarkable fortitude, represented by the following benefits.

• You have advantage on saving throws against being poisoned, and you
  have resistance to poison damage.


• You are immune to disease.


• You don’t need to eat, drink, or breathe.


• You don’t need to sleep and don’t suffer the effects of exhaustion
  due to lack of rest, and magic can’t put you to sleep.

and you'd replace the header with:

Skeletal Resilience

Your undead physiology gives you remarkable fortitude, represented by the following benefits.

There's a separate question here, which is that your player wants a race that "basically can't die". This is sort of unusual but honestly I think it's fine to let him have this. Add a feature where, when he's reduced to 0 hit points, he collapses into bones and then the bones reform after twelve hours. Being out of the story for twelve hours will hopefully be enough of a penalty that he won't start getting himself killed for fun.

If you like, you could implement this using the experimental rules for the revenant subrace from Unearthed Arcana.

(If you were planning to run a game where player characters die frequently, the "basically can't die" ability would be too good and you'd have to disallow it. Many DMs will run the whole adventure and not have anybody die, so in most games the fact that he can't die shouldn't matter too much.)

You've told us that your boyfriend actually wants to be a skeletal necromancer. This is likely to become be a problem.

For the first four levels, a "necromancer" is just a wizard -- wizards don't learn any good necromancy until level five.

After that point, your necromancer has a bunch of skeletons following him around who he can send into battle. This can produce balance issues (is the necromancer character more powerful than other characters because he's got all those skeletal allies?).

It also can produce alignment issues. Is your skeletal necromancer a good person? Is he a person who has friends? Does he treat his friends with respect? Or is he all "FOOLISH MORTAL FLESH-WORMS, I WILL DEVOUR YOUR SOULS" all the time?

In most worlds, animating skeletons is evil and only bad people do it. Most players who announce that they want to play a necromancer are intending to play an evil character.

So now you have a really hard question: if your skeletal necromancer is a bad person, why are the other adventurers hanging out with him? Why do they think he's their friend?

If you're sitting in the DM seat and you're like "yeah, your characters all have to be friends because that's the only way we have a D&D game", then you need to have all the players on board with their characters actually being friends with each other.

The closest thing to what you are describing is a lich

A lich is a powerful wizard that has embraced undeath, the details of which are described in the Monster Manual:

A lich is created by an arcane ritual that traps the wizard’s soul within a phylactery. Doing so binds the soul to the mortal world, preventing it from traveling to the Outer Planes after death.

In this case, the phylactery is the analog for a horcrux, as when the lich's body dies, it receives a new one since the soul is stored in the phylactery:

When a lich’s body is broken by accident or assault, the will and mind of the lich drains from it, leaving only a lifeless corpse behind. Within days, a new body reforms next to the lich’s phylactery, coalescing out of glowing smoke that issues from the device. Because the destruction of its phylactery means the possibility of eternal death, a lich usually keeps its phylactery in a hidden, well-guarded location.

However, as of yet, there are no rules for how a Player Character can become a lich, and certainly they wouldn't be able to start as one, since the creation and use of a phylactery involves the imprisonment spell (something you would only get access to at level 17).

The best alternative for PCs

The closest alternative available to Player Characters is the clone spell:

This spell grows an inert duplicate of a living creature as a safeguard against death...

At any time after the clone matures, if the original creature dies, its soul transfers to the clone, provided that the soul is free and willing to return.

Creating such a clone would allow a character to be safeguarded against death in a similar way to how Voldemort was safeguarded as a result of the horcruxes. This would be something a Player Character could have access to as a level 15 or higher wizard.

What about the skeleton?

No official races resemble skeletons, but you could get a more "undead" feel via some re-skinning of existing races.

"Basically can't die" is a danger flag here.

This is him asking for a major power-up that will break the assumptions of the gameworld. Necromancers are wizards. They're pretty fragile, especially at early level. Even if he was "just like a wizard, but he can't die" that would be severely unbalanced.

"Skeleton walking around" is potentially also a problem.

Most adventures assume that the party is composed of a fairly normal mix or accepted PC races. People might be biased about them, but no one's going to freak out when they walk into town just by looking at them. If he's a necromantic skeleton, those people would rightfully freak (unless "necromantic skeleton" was an established PC race that people had already grown accustomed to). If you let him play this, then you're going to be putting yourself in the position of constantly either leaving him out of certain parts of the content (because he's hanging around outside of town while the rest of the party actually does stuff) or coming up with vague justifications for why it's not a problem when it obviously should be. Admittedly, if your world is one in which necromantic skeletons are a normal part of society, this is not so much of a problem. Also, once he gets a few levels on him, a wizard with the right spells can mostly handle this. In a normal fantasy world, though, with a level 1 wizard, this would be an issue unless you as the DM put something in place to make it not an issue.

Now, if he wants to become that, then that's cool. The thing he's describing is a lich, and turning into one generally involves both being deeply evil and binding yourself in servitude to dread masters, but it will totally make you into a necromantic skeleton who's ridiculously hard to kill. It's also the sort of thing I wouldn't even start trying to do until lvl 15 or so. Up to you whether you offer him any less onerous ways of getting what he wants.

...though it'll still give you issues with the townsfolk.

Let me make this clear off the bat: you are in control of the game.

Can your boyfriend be a immortal skeleton throwing fireballs out of his eye sockets? Sure he can.

Will books give you a definitive solution? Nope.

Options

You can have a look at what a Lich is, but this makes him necessarily evil, so you will have to spin this off if that is not the idea.

You can have a look at skeleton creatures, but those are not immortal, so you do have to work out what differs him normal ones.

Liches actually have an artifact holding their soul, pretty much like Voldemort, only he breaks it apart and scatters it around. I don't know any supplement that would give you mechanics for that.

The system is only but a suggestion

I might come in playing the devil's advocate here, but sometimes following RAW is more exerting than things getting out of hand for not following it.

A ex-girlfriend of mine, for instance, found in her first DMing endeavour that for her game group it made more sense to make things up as she went, following RAW for basic mechanics, rather than strictly following rules.

In my group, people are more munchkins and they will create overpowered situations if I do not keep them in check. I would never have guessed DMing could be so freeform, in terms of dynamics, until I met her group.

So if you feel like your adventure is high fantasy enough, you could come up with some achievable way for player to get Voldemort-y.

So feel out your group. Do these people know the system well? Up to what level have they played before? Are they likely to go online searching for complements to find just the gear to make them unbeatable, or more like "I want a magic sword"?

The less munchkins player are, the easier it gets make it as you go.

World Consistency

How hard could it be... Immortality.

If a character could really be immortal in your world, its safe to assume someone would have achieved it already, right? It would be quite a popular thing to go after.

As others have mentioned becoming a Lich is supposed to be a really hard process, and it takes huge maintenance to keep.

Immortality, or something close to it, comes with great price, in nearly any story you pick to base yourself off, and sometimes it just ain't worth it in the end.

Anyhow, granting a player such power at level 1 is way out there. Voldemort is certainly not a level 1 character; your boyfriend's character right now is at most a Tom Riddle learning his first spells.

Can you make this work?

Conflicting groups are a challenge even for seasoned DMs.

What is the nature of your player's desire? It could be very hard to manage a group consisting of 4 Good clerics and a evil murdering skeleton. Aligning expectations here is quite important; if immortality is on the table, it could be the reason for the whole campaign.

Perhaps seeking the tree of life is the quest objective. One will try to use it to heal the village, another wants to study the arcane sigils on it, maybe someone wants to communicate with a dear one through a tea from the tree leaves. Your boyfriend wants to go immortal.

Scale the power up

Since you mention concerns over people getting OP: If someone is too powerful, immortal, or say, gets to convince someone they are actually a circus bear with a Deception roll, throw everyone up a notch.

And let them use their unique OP powers. I had a table once get out of hand because I allowed some players to get demonic or angelic subraces; they had resistance to elements, stat increases, minor spells, you name it. The one that went with a dwarf had nightvision.

His life was miserable in combat compared to others; next time he went unconscious, I gave him a power similar to Lucifer (in the TV series) where he could sense people and their intentions.

While the other players kept getting most of the fun in combat, I gave the dwarf power over all social interactions, something only he could do.

It can get out of hand and you end up with an anime table where everyone is throwing planetary nukes around, but as long as everyone's having fun...

You Rule

He's your SO so we all like to avoid conflicts, but if your campaign plan is not compatible with that level of power, he, more than anyone else, should be able to play along into the world you want to DM.

After all, role-playing games are a collective adventure, and you're playing as much as they are.