Khajiit or Argonian. You're an outsider to Skyrim, abused by the Empire and distrusted by all. Seek your vengeance from the darkness. You are a craftsman at heart, and death is a beautiful art you seek to master. Join the Thieves' Guild and The Dark Brotherhood, they will help you hone your deadly craft.
You travel fast and light. Start with a bow until you can replace it with Bound Bow. Enchanted clothing of your choice. Nightingale Armor once you earn it, though armor will not be necessary - you will not be seen, and will not be hit.
Distract your enemies and assassinate them before they even know you're there. Use one of your many invisibility potions to vanish into the shadows. Use a tanky follower to rush into battle ahead of you, allowing you to stay hidden. Summon Atronachs or Zombies to further distract your enemies. Summon your Bound Bow, apply poisons, and snipe away.
60/40/0 Magicka/Stamina/Health. Magicka to improve your summoning and stamina for those sniper shots. You shouldn't need additional health, your enemies will never see you coming. If you do happen to get injured, you will have plenty of healing potions to get out of a pickle.
Sneak - Grind up to Deadly Aim and Muffled Movement perks
Conjuration - Get the Bound Bow ASAP, add Mystic Binding and Soul Stealer
Progress Alchemy as much as you can, adding some skill points up the Poisoner tree.
Enchanting - Once you gain the Soul Stealer perk, you can begin enchanting. Focus on making money as you grind, then create a set of equipment to fortify conjuration, one for alchemy, and one for enchanting.
Pickpocket - Perks up to Poisoned. Once your poisons are potent, sneak in and plant them on enemies.
Speech - You're going to need more gold from merchants and more fences.
Use all that money to purchase property if playing Hearthfire - Windstad Manor is best for your Alchemist hideout. Cold and desolate, no one will come looking for a thief here. Grow a nice little garden, plant your greenhouse, build the alchemist tower, and stock your pond with fish for your potions.
You can dedicate 4 perk points to Illusion to gain Silent Casting, if you choose.
Expand your Conjuration up the Atronach or Necromancy arms of the skill tree. I selected Necromancy for more spooky flavor.
Add optional vampirism for additional stealthy fun in conjunction with The Dark Brotherhood.