Leveling Alchemy and Farming Gold

1. Get the Thief Stone blessing. Alchemy is a stealth skill and will progress 20% faster.
2. Create a house using the Hearthfire expansion. You will need to be Thane somewhere and have 5000 gold.
3. The house has an outdoors plot of land for growing herbs. Very recommended to also create a Greenhouse, too.
4. Grow Creep Cluster, Mora Tapinella, and Scaly Pholiota in equal amounts. Each harvest will give you three samples (not sure if Alchemy perks increase this).
   1. Grow the Creep Clusters in the individual planters. It makes them easier to harvest.
   2. For easier harvesting you can also hold your cursor in the space above the herb. The mess with the planter option is closer to the ground.
5. Go to the alchemist in Whiterun (or the Dark Brotherhood one mentioned below).
6. Wear all items with alchemy bonuses. Check the vendors for these (and barter items) every time you visit them.
7. Use all three herbs to make potions.
8. Wear all items with barter bonus.
9. Talk to the trainer, train any levels you can first. This will add to their gold. Remember if you leave the store, the gold will go away.
10. Sell everything you can.
11. Rinse and repeat. Do the max training every possible level.

The actual post from this link on reddit https://www.reddit.com/r/skyrim/comments/23w2jy/what_is_the_fastest_way_to_lvl_up_alchemy/ch14wwe/ is copied below in case it gets deleted in the future.

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Do you have Hearthfire?

If so, build a house or two (the Falkreath one is the easiest to get, but the other two don't take long either). Don't bother building anything beyond just the "small house" -- all the building materials are already on hand in the chest next to the drafting table. Then build the garden.

Plant five samples of Mora Tapinella, three samples of Scaly Pholiota, and three samples of Creep Cluster. There are 11 total plots so this will fit perfectly. Every three days, you'll get 15 samples of each ingredient. [Note: You must wait away from your garden in order for the plants to respawn!] If you have two or three houses set up, you can fast travel from one to the next, then wait a day or two and repeat the cycle. Keep going until you have ~200 of each -- it shouldn't take more than about 20-30 minutes depending on how many houses you have.

Now, go grab the Thief Stone near Riverwood, and sleep at an inn for the Well Rested bonus. Then travel to Arcadia's Cauldron in Whiterun -- or if you're in the Dark Brotherhood, go to Babette at the sanctuary instead. Pay Arcadia/Babette for five levels of Alchemy training, then make potions until you level up and can buy five more levels of training. Sell your potions to Arcadia/Babette to finance your skill training -- they'll use the money you gave them to buy back your potions! Just keep alternating between crafting potions, leveling up, and paying for training. It's a very quick process -- less than five minutes to use up all 200 ingredients. Babette can train you up to level 90, and Arcadia to level 75.

If you run out of ingredients and aren't at level 100 yet, then you should be close enough that a skill book or two will take you the rest of the way. All told, you should be able to get from 15 Alchemy to 100 in little over an hour -- faster if you've already got your house built and a bit of cash, and even faster still if you're already in the Brotherhood.

If you don't have Hearthfire, then you can still use the method of selling custom potions to pay for training at each level-up. Babette is still the better option, but as before, Arcadia works fine if you aren't in the Dark Brotherhood. Still use the Thief Stone + Well Rested combo too.

But instead of the Creep Cluster + Mora Tapinella + Scaly Pholiota combo, I suggest Blue Mountain Flower + Blue Butterfly Wing + Wheat. My favorite paths for collecting these ingredients are:

• Start at Honningbrew Meadery near Whiterun, and head North along the path. Grab Wheat from the farms you pass, along with the Blue Mountain Flowers and butterflies that litter the trail.
• Start at Windhelm Stables, and head East along the path. Again, plenty of Blue Mountain Flower on the trail, and several farms with nice Wheat stocks to hit as well. No butterflies though.

For that first route, the butterflies respawn each time you fast travel back to the meadery, so you can get those very quickly. Blue Mountain Flowers are also absurdly common. Basically from exiting Helgen, just follow the trail through Riverwood to the Honningbrew Meadery, North past Whiterun until you hit the snow. You'll pass dozens of samples. But these take a couple of weeks to respawn, so make sure you take every one you can find.

The Wheat is the least common, but you can find it at most farms across Skyrim and also at any apothecary. If you follow the route I listed, you should be able to get well over 50 samples very quickly. From there, just look for farms near any horse stable (every city has at least a couple of farms within eyesight of the gate).

Also, regardless of which of these two methods you use, be sure and pick up every new perk in the Alchemy tree (Alchemist x5, Physician, Benefactor, Poisoner) as soon as they become available. Also, wear any Fortify Alchemy gear you can find: Potion magnitude affects experience earned, so maximize your magnitude to maximize experience! The very first DB contract will net you Muiri's Ring, for +15% Fortify Alchemy. Between that ring and training to level 90 with Babette, the Brotherhood is definitely a nice home for budding alchemists.... And in addition to training you and buying all your potions, Babette also stocks some ingredients for you to use once you hit level 100.

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