I'm trying to understand the idea of putting alchemy as a 'stealth' skill. Is this just for convenience or is there some justification for it lore wise?

For example, most of the time I remember seeing altmer at the mage's guild being alchemists which gave me the impression that it requires some sort of magical talent, like enchanting. Has the lore for alchemist ever focused on anyone who is not some sort of mage or necromancer? Does anyone know how alchemy is supposed to work? Do you just mash the stuff up and put it in a bottle with some water?

In-game convenience. Alchemy doesn't require the use of magica, and poisons are something stealthy characters would use, hence it could belong in stealth.

As for how alchemy works, we've got a couple of http://imperial-library.info/content/oblivion-potion-recipes in lore. It indicates that there are specific proportions to the ingredients, and that sometimes the solvent does matter. Presumably, you can also eat the combinations as a solid for a similar effect.

everything has propertes, same as in real life. if you mix Aloe Vera with the right stuff, you get sunburn relief. same as alchemy. no magic to it.

Well, there is magic involved, but it's innate in the substances, and has nothing to do with the alchemist.

It's an arbitrary reassignment so that each archetype has one crafting skill and five gameplay skills. Alchemy was always halfway between magic and stealth anyways, since it was a major skill of nightblades and assassins. Makes more sense than Enchanting being under the Thief.