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. 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.
(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.)