First off, Archers need to put NOTHING into Stamina or Magicka, as you never use Stamina or Magicka. If you can't get a slow-mow snipe shot off in the time 100 Stamina runs out, you're doing it wrong.
Health is the only thing you need to increase.
You want to level up Light Armor, Sneak, Archery, and Smithing (imo) so you can craft the best bow in the game, Daedric. Nightingale bow is bad compared to the Daedric, especially if you enchant it.
Race doesn't matter at all, unless you want constant Night Eye, which is REALLY helpful.
I don't recommend using Alchemy as a main skill since snipe shots kill in one shot 90% of the time, poisons are a joke. They do nothing, as people die before it takes effect.
Also, Illusion is pointless for a true archer as things simply die before you have any need to cast a spell if you're sneaking.