This build is based on the Battlemage class as seen in TES3: Morrowind.
The Battlemage is a heavily-armored spellcaster that uses Enchanting, Destruction and Conjuration magic to supplement melee damage. The schools of Restoration, Alteration, and Illusion come from potions and Alchemy.
The Battlemage can be any race, but is well suited to Altmer, Bretons, and Dunmer. The Lord Stone is chosen over the Apprentice and Atronach stone due to the way Spell Absorption interacts with conjured creatures in an unmodded game. If using mods, the Atronach stone is preferable.
Most attribute bonuses should be put into Health and Magicka; Stamina is unimportant unless you like to power attack a lot. Being able to make your own potions can mitigate gold costs until later in the game, when powerful restore potions are common in dungeons.
Combat style should be a mix of utilizing Conjuration in conjunction with Destruction while at a distance and then closing in with a war ax.
The Battlemage's gear should consist of heavy armor pre-enchanted with Fortify Magicka, Fortify Magicka Regen, and Fortify Destruction and/or Conjuration effects. Enchanting armor is less important, since much of those effects can be achieved through Alchemy.
If roleplaying or are keeping with the combat magic theme, shouts can be any damage-dealing or summoning shout, including Unrelenting Force, Breathe shouts, Storm Call, and Dragon summoning shouts.
Perk selections may be different if: 1) armor weight becomes an issue with heavier armor like Daedric or Dragonbone; 2) you choose to specialize in a particular element; 3) you decide to enchant your own armor instead of relying on loot and vendors.
Note: The Destruction perk "Intense Flames" can trigger a bug that can break the quest "Arniel's Endeavor." If you choose to spec in Fire damage, don't take the first level of that perk until you complete this quest.