Best way to raise alchemy in my experience is to make the most expensive (and thus giving most experience) potions you can with the ingredients you have. Salt Pile and Death Bell makes a Slow poison, and two of Blue Butterfly Wing, Blue Mountain Flower, Hanging Moss, Nightshade makes a Damage magicka regen poison. Both of those increase Alchemy pretty quickly and net a good profit (also improving Speech)

In my playthrough I'm picking pockets all the time, so I always have a ton of rings and necklaces to enchant with fortify sneak (for grand, greater and common souls) and waterbreathing (for lesser and petty souls). I've seen guides recommending smithing iron daggers and enchanting them. Just use whatever materials you can get without going out of your way too much imo.