What are Nazgul?
http://lotr.wikia.com/wiki/Nazg%C3%BBl

Black robe, black hood, and a one-handed sword. That's all the equipment I need. Of course a black horse, preferably Shadowmere would be awesome, and.... ice spells? What would you suggest?

I know of the Necromancer robes, but where are some basic black robes? And a basic black hood? I found a green one at Radiant Raiments, but black one. Also, do you think it'd be the best to use ice-magic for the Nazgul? I was thinking of enchanting my swords with ice magic and using an ice spell in my free-hand. Also, other than Shadowmere, what stables sell black horses?

If you're to make a Nazgul type of character, Shadowmere is the mount of choice.

A hoodless dragon mask (if you're on PC and have access to mods) can be a good addition -Nahkriin mask seems fitting. Otherwise, get rid of the black hood and just use the mask, as it is in the default game. Should fit nicely.

I don't think any spells in the free hand really suit, you're pretty much limited to a sword, black robes, black boots, a dragon priest mask and then enchanting and perhaps alchemy for poison? It doesn't seem all that feasible really, since you'll have to change a lot about the character to get by if you start adding Destruction for added damage and Alteration for defense and such.