Well this depends. The Question is, "How do I play a mage character?"
Short answer to this one is... you don't.
Look to the example(s) set in Oblivion... How did one play as a mage back then under the class system?
The first thing you have to understand is like what Farengar Secret-Fire said, that while he is a full-blown mage, "I'm sure there are others in Whiterun who practice as well". So this begs the next question, "What kind of mage are you?" The answer you choose has far reaching implications and each one is unique. Rather than cover them all, I'll talk about the main types:
Wizard - You're a clothy and magic of any kind is your bread and butter. For this reason, "The Apprentice" is most likely the stone for you. As any serious Mage requires fast Magicka Regeneration, everything else comes secondary. You could also take the Mage stone and mod your game to give the +50 Magicka bonus found in Oblivion verison of the "Mage Birthsign".
Battlemage/Spellsword/Nightbade/Crusader - This is what I usually like to play as. You have Armor to re-enforce your lack of toughness, and can use Illusion or Sneak to improve your chances of coming out on top of a major conflict. While at the same time, you have plenty of Magicka and Skill with it to fall back on when your Melee abilites falter or become useless.
The above classes are the best in the game IMO. Because let's say you're up against an opponent who is virtually IMMUNE TO MAGIC DAMAGE such as, "The Ebony Warrior". No way in hell you're going to be able to take him down with just Magicka alone, no matter the regen rate. You are going to need to be able to do lots of physical damage too.
Forget summoning things, this will seem like a very good idea at first but can bite you later, due to your constant "need" for someone else to do your fighting for you. The summons can always be banished or could possibly be turned against you. Illusion is the same. Not everything can be Paralyzed, although that's a nice trick. And those other Illusion spells will only work on NPC's up to a certain lvl and then they become too strong for them.
The ultimate answer to this question is both. You need to be skilled and adapted to BOTH Melee and Magic Combat. And unlike Oblivion, I think Skyrim gives players too much Freedom and doesn't have enough structure. And if you think this is something you won't need, then when your Mage in Robes w/ no armor whatsoever goes throwing lightening at some Redguard and then almost immediately gets cut in half by his Claymore of Doom, you might think differently. Maybe.
Hope that helps.