"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.