Alchemy is the most tedious skill to level up. Not only do you have to find the right ingredients to even be able to make a potion, but then every potion you make contributes very little towards leveling up. However, the leveling of alchemy is very profitable as some potions sell for quite a bit of gold.
The way I usually level alchemy is that I always pick anything I can whilst I walk around the countryside. Ingredients at alchemy vendors are pretty cheap, so I usually also just buy every ingredient that have for sale.
The next thing you will have to do is eat one of every ingredient that has unknown effects. To start with you will only be able to discover the top right effect of an ingredient, but a series of perks in the alchemy perk tree lets you discover more and more effects of an ingredient. After this you should always look at the potions you want most and make them first, the just go down the list and make as many potions as possible. You might also want to hold off using certain ingredients that are used to fortify smithing, enchanting and other useful effects such as invisibility and fortifications of certain skills until you have reached max level with alchemy so that those potions are stronger.