1. Make sure you buy Honeyside and bought both the alchemy lab, garden, and porch upgrades. Being in the Thieves Guild probably helps too 2. Make sure you've entered the college of Winterhold. Becoming Arch Mage probably makes this easier as well. 3. Make sure you have at least 1 perk in Speech
Now, the garden for Honeyside has alchemy ingredients that respawn, and the porch upgrade has barrels that have certain alchemy ingredients that I believe respawn as well, provided you don't put anything in them. When you buy the alchemy lab for Honeyside, it includes a large amount of ingredients and a few recipes laying around.
The second floor of the Hall of Countenance at the College has an alchemy room full of ingredients, all of which respawn.
The reason for the speech perk is so you can buy recipes and ingredients for cheaper
Also, discovering new effects from eating an ingredient does increase the skill a small amount, so if you have the perk that causes you to learn more then one, that'll give you a small boost as well
"You're dead if you aim only for kids. Adults are only kids grown up, anyway." - Walt Disney
Salt Pile + Deathbell yields a decently value Slow potion. Deathbells are abundant around Morthal and you can find Salt Piles most anywhere.
One of the best ways to level alchemy is to purchase all ingredients from a shop that are less than 20g. Make all the potions that you can and sell them back to the shop owner. Go to another shop (or save, attack the shop owner, and reload the save to reset their inventory and gold) then repeat.
Remember that you can continue to sell items to merchants for 0 gold to increase your speech rating. Once you obtain the Merchant Perk you can find new items in the alchemy shops and will have an easier time selling off your myriad of potions to blacksmiths, flechers, general stores, College members, etc.
Even a man who is pure in heart and says his prayers by night, may become a wolf when the wolfbane blooms and the autumn moon is bright. The Wolf Man - (1941)