I can't play anything but archers in Skyrim. If I try to play something else, even an interesting build like a spellsword, all I can think about is how much more fun I would be having were I playing an archer. So with that out of the way, let's try to think of as many different archer builds as possible. Here are a few popular ones:

Archer
Archery, One Handed, Block, Light Armor

Hunter
Archery, Alchemy, Smithing, Light Armor

Assassin
Archery, Sneak, Light Armor, and thieving skills.

Ranger
Archery, Alchemy, Sneak, Light Armor, Restoration

Arcane Archer
Archery, Conjuration, Alteration, Enchanting

If you can think of an interesting build that isn't listed here, .

For a straightforward Archer I'd actually go for Archery and Sneak, and then Light Armour and Block. Take advantage of sneak for silent kills from distance, and the awesome perk to assist that from the Sneak Tree. The Block Tree will fix you to efficiently butt people with your bow if they get too close, and give you vital seconds to run away.

Orc hunter
Archery, sneak, two handed but only with an axe or warhammer, one handed but only with a knife, light armor

No magic, at all! You can buy potions.

Frontline Archer
Tanking enemy damage like a boss. Don't knock it until you try it.
Archery, Heavy Armor, One-Handed, Smithing. Compliment with Enchantming, Alchemy and Restoration.

My Orc Ranger / Thief

Archery
Light Armor
one handed (war axes)
alchemy
smithing
sneak

Focusing on health and stamina, 3 health, one stamina. So I can actually defend myself when the [censored] hits the fan.

of course pickpocket and speech are lightly touched on too, and smithing is done, because apparently orcs learn to smith in the womb, lol.

Does the power bash stagger for longer than the standard bow bash?

Power Bash is more of an attacking sweep. But I think it might.

I don't like using sneak for my archers. It makes combat boring IMO.

My current build is:
Archery
Light armor
Alchemy

I will probably start working on my One handed and block skills once these are high enough.
I put every point on Stamina.

EVERY POINT?! That means at level 50, you would still have 100 health, and get one shot by just about everything.....

I know. I like glass cannon type characters.

I don't like sneak either. But why level one handed and block? It was because of the lack of skills for a sneakless archer that I decided to use a long bow exclusively and level smithing so I could upgrade it a bit, as well as my armor (pretty ironic that taking smithing will result in a more difficult game ). Together with lockpicking and speech, that should be enough to hit the soft level cap, at which point I'll retire the character.

Love the thread, I'll post mine when I get home. (on my iPhone)

Archer Thief- (low health, low magicka, extremely high stamina)
archery
sneak
pickpocket
lockpick
speech
light armor

my lvl 38 charater has 160 health 100 magicka and 410 stamina, it works out pretty well