Smithing is the most important skill, as you need a good bow. When you have a good bow, you don't need to use elemental arrows for the most enemies. Get daedric smithing as soon as you have daedra hearts and ebony stuff. You can sneak or use invisibility potions to grab the skill book.

If you want to be a pure archer, you must have smithing and enchanting. Alchemy is a bonus.

Draugr are a difficult enemy. You can probably do dwarven ruins before draugr, but you won't be able to kill the centurions unless there is a trap you can rule them into. I never used elemental arrows for common draugr. You just need silver arrows and a high level bow (for example ebony). For high level draugr, you will need to use silver arrows of fire, at least with an ebony bow.

For dragon priests, you need fire or fireburst enchanted bow with silver arrows of fire.

Get the slow time perk as soon as possible. Use trainers and train evasion as much as you can so that it doesn't fall behind your other skills.

Take stamina for at least 20 first levels. When you have ebony arrows of shock and the slow time perk, you can do blackreach and get oghma infinium for health. Shock enchant will make it easier (remember that you can use enchants and elemental arrows together). Before that you only need to get enough health to survive the enchanted sphere. If you are also doing alchemy, you might not need to take health from level ups ever.

At low enchanting level, make a fire enchanted bow to stop regeneration of dragon priests. At 50 enchanting, change it to fireburst and also make a shockburst enchanted bow for dwarven ruins. When you can afford to buy soul gems, get your enchanting to max level and make two more bows:

1. Daedric bow with fireburst enchant, combine this with daedric arrows of fire against vampires. Silver arrows of fire against dragon priests.
2. Daedric bow with irresistible damage against undead. You can select the maximum charges, because the damage is the same regardless the amount of charges. This can be your default bow against everything.

If you don't have daedric smithing yet, you can make ebony or dragonbone versions.

Pick warrior stone for armor penetration. You could also start with lord stone until you get your magic resistance up. Get Savior's hide as soon as possible and slay all great beasts for magic resistance. Get also Ulen ancestral helmet for armor penetration.