So I want to make a character that goes for all shouts, and I was wondering what the most efficient way to make this character would be. Obviously a lot of dungeon crawling will be involved, so should I go sneak, or what?

All opinions welcome.

Vampire mage with illusion as the main class is what I'm running with right now.

Sneak is slow. If you want fast and efficient, be a tank and set the difficulty to a minimum.

I'd say go the opposite of sneak. If you go with sneak its going to take longer to get through the dungeon because you move slower ( especially when you are a low levels as you will get detected easier and have to stop). However, if you go with like a two handed (or even one handed) warrior with heavy armour and restoration (so basically a paladin) and a crap ton of health you could easily cut through a dungeon really fast. If you don't want to be the good guy replace restoration with conjuration and be the evil overlord who uses the bodies of his enemies as slaves (my current character ), or you could use both restoration and conjuration, either way works.

Fire and runes. Purge the undead with flame and burn through them all like a walking flamethrower

Sanguine Rose and Eola with the Staff of Revenants. Doesn't mattaer what build you are with those, you can pretty much clear anything.

A Two hander wielding, heavily Armored cook, who has a passion for collecting Leeks and salt piles to make Beef Stew, Vegetable Soup, and Venison Stew.

Heavy Armor, dual-wield swords, Smithing and Enchanting worked for me. It'll take a while to get all that up, but once you do you'll be one-shotting even Death Overlords with a Power Attack.

Sneak isn't bad either because you can one-shot the Draugr before they even get out of their burial niches, but you can't just run through.