This is quite a long process...
First you need to go to the alchemy shop in White Run and you will need a do not delete chest to do this.
Skip the first version if you don't have very good pickpocketing.... Now to start ask her to train you one time, not all five levels and when she does she will take around 100 to 300 gold. When she does, sneak around the counter and pickpocket her to get the money back. Then repeat the process until you have leveled up 5 times. Then wait until you have leveled up or start off version 2 of leveling your alchemy, while you wait to level up your main levels.
To start, when you walk in, go behind the counter with a platter (platters work the best when it comes to wall glitching).
Walk towards the wall where the entrance is but behind the counter, then hold the platter in front of you, then run towards the platter and if you keep sprinting you will notice that your running through the wall.
Once you get through, you will start to fall. Then quickly look down and drop the do not delete chest.
Then if you look around you will be able to see two chests. One is another do not delete.
Look down then quickly jump, tap a about three times very fast and go into you inventory and drop it again.
Then you will slowly work your way up. Once you get to the chest, take all the ingredients and just fall down.
Then you will be teleported back up to the floor and you can make a bunch of potions and sell them to the old lady. Then save, kill her, and reload the save, then ask what she wants for sale.
Just look at her inventory, jump back onto the chest through the wall, and search the chest and take all the ingredients. While you're selling and making potions, you're leveling up your skills.
Then once you level up again, ask her to train you in alchemy.
[more cheats for The Elder Scrolls V: Skyrim here]