So I wanna make a character that uses primarily illusion and archery. What other skills should I look into?

Leverage sneak and one-handed. By the time you get Invisibility, you'll be backstabbing more than you're shooting.

Archery will still be useful for dragon fights.

With illusion as your main skill you can just drink coffee and let others fight eachother.

Around Illusion and Archery , i would use Enchanting-Smithing-Alteration (or Light armour ) as the 5 main skills , with a few perks in the right side of the Sneak tree for stealth damage multiplier with bows .

With invisibility and muffle spells of illusion , you won't need more than 1/5 perk in sneak , you will be undetectable through magic . Enchanting so you can craft proper clothes or armour with Illusion/Alteration cost reductions or magicka/magicka regen increases and Smithing for your bows and armours if you use them .

Sneak, Enchanting, Light Armor, Conjuration - bound bow is one of the best bows, if not the best bow, in game. Summons will keep melee off of you on the occasions they live through your sneak attack shots.

Bound bow is a just a basic Daedric bow i think , but that shoots daedric arrows , that's where its interest is , powerful arrows for free . But If you are strong in Smithing/Alchemy , you can craft bows with 400+ damage and still use high end arrow like glass/daedric .

I think the strong point of bound weapons is in the early game , it gives you very powerful weapons compared to the steel/elven stuffs that you can find , but past level 30 and providing you use smithing , you can easily make much better bows , but i agree Conjuration is a very nice skill , you can't go wrong with it

Bow draw speed is based on their weight though, and the bound bow is weightless. At least from what I've heard, haven't tested it in game yet although it'd be pretty simple to check.
I used a bound bow up to level 53 on my last character and had no issues on expert difficulty. I'm sure a legendary daedric would be better on expert or anything made with stacked smithing bonuses really, but the bound bow is still quite powerful. The other bound weapons, not so much.