In my experience, you need around 50 Alchemy and 2/5 Alchemist to start seeing an effect. The process itself is very simple, and there are plenty of videos on YouTube demonstrating it. Pretty much as follows though:
1. Acquire some Fortify Alchemy gear; more pieces is better, and higher value per piece is also better
2. Acqire ingedients for around 10 Fortify Restoration potions.
3. Make a potion while wearing the Fortify Alchemy gear.
4. Drink the potion, then immediately re-equip the Fortify Alchemy gear.
5. Make a second potion while wearing the re-equipped gear.
6. Drink the second potion, then immediately re-equip the Fortify Alchemy gear.
7. Repeat steps 5-6 until desired magnitude is reached.
You will need the benefactor perk as well as 2/5 in alchemist if you want it to go faster also get enchanting up to 50 and grab the insightful enchanter perk. Ingredients for restoration potions are abecean longfin, cyrodilic spadetail, and salt pile. The easiest places to find the fish are the lake outside riften and the lake in falkreath. The salt piles can be found in the barrels all over riften.