This is untested and unfinished, so please feel free to criticise.
Alchemists have the same XP requirements, Hit Points, weapon and armor restrictions, and attacks as Magic-Users. Their Prime Requisite is Intelligence.
They start with a Saving Throw of 15 (like Magic-Users), but gain +2 vs poison rather than vs spells (edit: if using Fortitude, Reflex and Will, they should have a good Fortitude save. I find Basic/AD&D-style saves quite difficult, and I'll do them in the next draft).
They start play with two potions, which are rolled randomly as if generating treasure.
In areas with the proper facilities (usually large towns and cities), they may make potions when not adventuring. It costs them 500gp to make a potion of their choice, or 300gp to make a potion which is randomly rolled.
Their bonus against poison increases as they gain levels: +3 at level 2, +4 at level 3, and so on.
(What else happens as they go up in levels? I'd say that making potions becomes cheaper, but adventurers should have more gold anyway. Maybe they gain access to poison, or the ability to create other types of magic item).
Optionally, they can create and start with miscellaneous items as well as potions.