It depends very much on who you're covering, and what you're doing with it.
At least in the USA, you don't have to do anything other than notify the artist you're covering them, they can't stop you. You may or may not have to license it to release it publicly though.
If it's a big name covered under ASCAP/BMI/etc. then you can likely license it under
Harry Fox Agency's Songfile service. The annoying bit is that you have to pre-guess how many streams/purchases etc. you expect so you can license the right amount.
It gets stickier, too, in that some of the big sites have their
own agreements so that you can do covers directly on the site without specifically licensing them, but it depends on what countries it's getting distributed to.
I use Routenote for my distribution and they
have a whole FAQ about it.
On the other hand, if it's a small artist like most DungeonSynth stuff is gonna be, most likely the only thing you have to do is email them and ask permission.
![:)](//storage.proboards.com/forum/images/smiley/smiley.png)
Personally, I would be thrilled to have someone covering my stuff and I wouldn't expect compulsory licenses or anything.
If you're not charging money for it and you're just sticking it up on Bandcamp or whatever, basically no one is going to go after you for it, AFAIK. If you start distributing to the "real" platforms then you have to start handling at least some of the legal stuff around it.