Quoting user TodKarlsen on elderscrolls.fandom.com:
- Bring the ingredients for your favorite Fortify Restoration potion to your Alchemy Lab. Salt Pile is an obvous choice, then the other ingredient is is a matter of what you have a lot of: Abecean Longfin, Cyrodilic Spadetail, Small Antlers, or Small Pearl.
- You'll also want to have the ingredients for Fortify Enchanting and Fortify Smithing potions.
Fortify Enchanting ingredients:
- Blue Butterfly Wing
- Hagraven Claw
- Spriggan Sap
- Ancestor Moth Wing (Dawnguard)
- Chaurus Hunter Antennae (Dawnguard)
- Spawn Ash (Dragonborn)
Fortify Smithing ingredients:
- Glowing Mushroom
- Sabre Cat Tooth
- Spriggan Sap
- Don your best Fortify Alchemy gear (5 pieces if you use Circlet & Falmer Helmet).
- Craft your first Potion of Fortify Restoration @ 147% (yours might be lower, my Alchemy skill is at level 100).
- Drink it, but before exiting the Items menu, go to the Apparel page and doff and re-don one piece of your Alchemy gear. This makes no sense, but it's necessary to bypass a cap which only allows the potions to be 363% stronger.
- Immediately go to the Alchemy lab and craft the 2nd potion @ 363%, exit alchemy and drink the 363 potion. Go to the Apparel page and doff and re-don one piece of your Alchemy gear.
- 3rd potion @ 427%; 4th potion @ 1165%; 5th @ 4428%; 6th @ 41573%; 7th @ 3075947%; and so on, but you risk crashing the program.
- Then make your Fortify Enchanting and Fortify Smithing potions, and whatever other potions you want insanely powerful.
Check the thread linked above for more information on using the exploit (note, for example, that it has been patched by the Unofficial Skyrim Patch (which got 'unpatched' by this mod), and that the values depend on your skills, and other variables).
Here's an amusing video by The Spiffing Brit showing you the same process, along with a few more exploits.