One more thing, I'm stuck on the race
My alchemists tend to be Bretons but as you are going for an assassin I would say Khajiit (+5 Alchemy and one handed +10 Sneak)
Alchemy, Sneak and Archery. All are major aspects of my current character.
Some people talk down the skill of alchemy, but those are ones that don't employ it very long. It is arguably the most powerful, if not overpowered skill in the game. It is, without question, the most versatile skill of them all.
http://www.uesp.net/wiki/Skyrim:Salmon_Roe link about creating the most expensive poitin in the game.
http://skyrim.melian.cc/?cmd=cmdSkyrimAlchemyWizard is actually cheating but it is quite useful nonetheless.
I agree with archery. With the right perks, along with sneak, you're talking one hit kills.