What are the fastest way to level up Enchanting and Alchemy?

To level up enchanting to 100, you need to enchant 310 items. To level up alchemy to 100, you need to make 550k worth of potions.

To level up enchanting to 100, you need to enchant 310 items. You need to disenchant at least one item. Then you need 310 soul gems, and 310 iron daggers or gold rings or something that can be enchanted. You can do it bit by bit, or if you have enough materials, all at once.

You can make a lot of money if you have the right enchantments. In early game, damage stamina makes good coin. Then paralysis and absorb health is slightly better. If you are over level 24, you can find or buy a weapon with Banish enchantment. They make a LOT of money. Enchanting jewelry is generally less profitable. But fortify sneak is good.

To level up alchemy, you need to make 550k worth of potions. Stronger potions make more money, so you want to make potions as strong as possible. You can make stronger potions by taking perks in the alchemy tree and wearing enchanted items. If you want to make it easy, I would recommend delaying alchemy as much as possible and gather as much materials before you start making potions. Get training with Arcadia with all that money you made with enchanting. Having enchanted items is very useful for making stronger potions. You can find a circlet of alchemy, necklace of alchemy, and gauntlets of alchemy. If you can enchant well, then you can enchant a whole set yourself.

Some potions are worth more money than others. Potons with Damage Magicka Regen, Paralysis, Invisibility, Slow effects fetch good coin. Most expensive effect is 300 second Fortify Health made with Giant Toes. You want to collect all ingredients, but there are some rather common ingredients that make nice expensive potions.

Blue Mountain Flower + Blue Butterfly Wing = Damage Magicka Regen
Blue Mountain Flower + Blue Butterfly Wing + Giant Toe = big money
Bear Claws + Hanging Moss + Giant Toe = big money
Chaurus Eggs + Luna Moth Wing = Invisibility
Salt + Deathbell = Slow
Canis Root + Swamp Funal Pod = Paralysis

If you are going to have a big enchanting session or a big potion making session, make sure to take advantage of guardian stones and sleep bonus. You are going to level up 35% faster. It makes a big difference.

http://www.uesp.net/wiki/Skyrim:Leveling