The way to improve alchemy is by creating high value potions, if your serious improving the skill will help you brew more valuable potions.
The cheapest (in ingredients) high in value is a slow poison by mixing a deathbell and a salt pile but you may want to add a third ingredient to further increase the value even if it's a positive effect it'll increase the value

I made 2 potion of 2009 value and it increased my alchemy from 89 to 90

The most valuable potion a character can make is from Giant's Toe + Blue Butterfly Wing + Blue Mountain Flower.

Blue Butterfly Wing + Blue Mountain Flower + Giant's Toe = FortConj (75) | DamMagReg (265) | DamStam (43) | FortHp (485) | Value: 868

The higher the base value (base means minus the effects of speech/haggling) the more skill you will gain. With higher alchemy, the effects will be stronger, thus the base value will be higher. So with higher alchemy and alchemy perks, you will gain more skill per potion.

Regardless, the above potion is the best you can make, so making lots of it will improve your skill faster than any other.

Wearing +Alchemy gear and taking the first perk(s) also makes it go faster, since the enhanced effects make the potions worth more.