It should be fairly simple, honestly.
Race is really irrelevant but Altmer or Breton would be a good choice. Altmer will give you an additional 50 Magicka upon start and a good racial. Breton will give you a good racial as well, a good passive ability, and it will also give you the spell "Conjure Familiar"... which is why a Pure Mage would be simple.
You're going to want to invest heavily in Conjuration at the start, or Illusion. Conjuration will let you summon an ally, essentially a tank, that can do the big damage for you as well as take the hits that would otherwise cripple you. Illusion can be used paired with sneak to make enemies fight each other until you're left with a 1-on-1 scenario. You can mix Conjuration into that by using necromancy to resurrect the people your Illusioned target just killed and have them fight again. It gets very amusing sometimes.
The issue is that spells don't scale as well as weapons in the later levels.
Translation: You won't do a lot of damage towards the end.
You can, especially if you choose Breton, pretty much invest all of your points into Magicka but I'd obviously split it between that and health, should you be caught off guard.
I also recommend enchanting your gear to the point that your highest Magicka cost spells (probably Conjuration) cost 0 to cast, as well as probably Fortify Destruction to do more damage.
Also, if you invest in the Bound Bow early on, it's almost on par with a Daedric bow. Still technically magic, but far more lethal.