First of all, a lot of things have changed since the video. I've revamped the character, though he has the same goals. Every Enchantment, Spell Tome, and Shout. Used to, all Magic Perks was a thing but later decided it wasn't necessary. If you have extra perk points and want to take the other Magic perks, go for it.
This character is an Altmer Pure Mage, meaning no armor whatsoever except through Alteration (or The Lord Stone if you decide to use it). Here's the gear you will want to go for as far as end game. When it comes to Fortify Destruction and Restoration however, you'll want to try to shoot for at least 29% with both on 3 pieces of gear (Circlet, Ring, Necklace) because the torso is going to be the Archmage Robes. Reason being is that it'll be enough to eliminate Destruction and Restoration cost entirely.
Circlet of Stardust (Silver & Moonstone Circlet - Fortify Destruction & Restoration)
Amulet of Stardust (Bonehawk Amulet - Fortify Destruction & Restoration)
Gloves of Stardust (Thalmor Gloves - Fortify Alchemy & Fortify Magicka)
Ring of Stardust (Bone Hawk Ring or Gold Diamond Ring - Fortify Destruction & Restoration)
Boots of Stardust (Thalmor Boots - Fire Resistance & Frost Resistance)
The Main Storyline
College of Winterhold Storyline
The Book of Love
* - Mainly doing those quest for money. Though, selling potions will be the main money-maker.
He does have a weapon that he uses, and it's the only one (besides any staff).
Starblade (Stalhrim Dagger - Chaos Damage & Frost Damage)
The Fortify Destruction skill on all your gear will give this weapon a permanent charge.
Obviously, it's recommended you have the best Alchemy gear for this character. You'll also want Azhidal's armor & Seeker of Sorcery for enchanting bonuses along with the best Enchanting potion you can make.
If you have any questions, please contact me via Twitter @InfiniteOni.
Note: I know in the picture he's wearing Stalhrim Gauntlets and Boots. I thought it was a great aesthetic for the character. If you would like to go Heavy Armor and use them, you can. Just don't bother taking the Mage Armor Perks, and take a few perks into Smithing (Steel, Arcane, Dwarven, Orcish, & Ebony). You'll also want to use the best Smithing Gear and Potions to get the armor cap.