I'm only level 20 in it and it seems like it takes 30+ successful potions just to reach one level. Am I missing something?

Here's a cool tip :

When you kill dragons, you get Dragon Bones & Scales. Keep them. Go to Whiterun and go to the Alchemy Shop. Speak with the owner and train your alchemy (you can train it 5 times per level). Then sell her your Dragon Bones & Scales so you get your money back. You can also mix a lot of potions and sell them to her, so your Alchemy training comes VERY CHEAP

You can make armor out of the dragon parts so think before selling them as you should probably have enough gold.

As you sell potions you don't need you should earn money as long as you don't buy expensive ingredients.

Don't you need to have 100 smithing to make dragon armor, though?