I know this sort of thing has been done before, but here's my own take. It's a pretty long read, though - you have been warned.
Race: Altmer. 5 free levels of Magicka is always nice, Altmer come with a suite of magic skill bonuses, and the speedy regen power could be of use.
Stone: Warrior in the early levels to ensure that Archery advances quickly. Later, switching to the Lord stone is always a solid choice. Stay away from the Atronach due to the Conjuration/absorb bug, which I assume affects bound weapons as well as summoned creatures. Anyone have any other ideas here?
Attribute allotment: Magicka is somewhat important to this build, though the Arcane Archer doesn't do a lot of casting in combat. The Altmer racial bonus means you don't have to invest in Magicka for several levels, giving you time to work up health - though i played carefully, archers can get by with a little less health than other classes. Still, it's a good idea to work this up to avoid getting one-shotted. Finally, there's Stamina - an important attribute for the class. Without Sneak (more on this later), an Arcane Archer needs to be able to kite, and Stamina keeps him running longer. It's also needed to pull off those emergency bow-bashes when an enemy gets too close.
In the end, a distribution of something like 2 / 3 / 2 for Magicka / Health / Stamina would probably be fine.
Skills:
Archery - the signature skill of the Arcane Archer, obviously.
Conjuration - Exclusively for the Bound Bow. Perk up the middle of the tree to add a few tasty effects to your ethereal Daedric weapon, like soul trap and banishing/turning. The only downside here is that the soul trap effect can lead to you accidentally trapping a petty soul in a grand gem, for instance. To avoid that, reserve Bound Bow for situations where you want to soul trap or banish/turn specific enemies. Otherwise, use your physical (enchanted) bows.
Destruction - Mostly for Rune spells, which make a nice, cheap trap for enemies once you get Rune Master and can cast them far away. Slap down a rune to draw out foes, and wait with an arrow at the ready. Destruction is also nice for those times when enemies get too close; you've whittled them down with arrows, but they still have some health left. Out comes Flames, Firebolt, or Fireball. Think of Destruction as your melee backup - kind of like a dagger. Not a major skill for the Arcane Archer, but it does have some utility if used carefully. My sample build, linked below, doesn't go too deeply into the perk tree for this school, but I can easily see grabbing dualcast along with augmented/intense flames. (No need for Impact here; Power Shot from the Archery tree should be giving you all the stagger you need.)
Alteration - Why carry around bulky armor when you can make your own out of thin air and good intentions? Needless to say, the ___flesh spells aren't as effective as constant armor, but as an archer you shouldn't be getting hit as much as melee types anyway. There's also some nice utility here - some of the light spells can be used as distractions, Detect Life is obviously useful, and eventually Paralyze becomes a nice crowd control spell. And the perks are constant, passive defense. (But avoid Atronach perks for the reason stated above.)
Enchanting - This skill simulates the Arcane Archer's ability to 'imbue' his arrows with spell effects. Be sure to disenchant early and often to get access to the standbys of fire/frost/shock damage, as well as other 'utility' effects for your weapons. Carry around several bows with different enchantments to match to the situation. And don't neglect enchantments for your clothes, hood, boots, gloves, ring, amulet, and what-have-you.
Skills Notable By Their Absence:
Sneak - Most people probably consider Sneak essential for an archer. But the fact is, perk out Sneak and you won't need an enchanted bow, Alteration, Destruction, or ... anything, really. You'll be making sneak attack after sneak attack all day long. This is fun for awhile, but eventually.... yawn. Ultimately, Sneak doesn't really fit with the typical Arcane Archer - but I could be persuaded to grab 3 perks in the tree, up to Deadly Aim, if I thought I could discipline myself to save sneak shots for those really critical situations. I'm a weak, weak man, though...
Illusion - Seems like a natural fit for an archer, who necessarily will want to keep crowds of enemies out of his face. But the Arcane Archer isn't a sneak archer (at best, he's minimally sneaky), and Illusion doesn't seem much fun to me. In my experience, the problem with Illusion (crowd-control spells) is that there's no middle ground. It either makes combat a non-issue, or it doesn't work at all. And if I'm Frenzy-ing my enemies all the time, what's left for me to shoot?
Smithing - A case could be made here to get up to 60 in Smithing and take Arcane Blacksmith, to improve enchanted gear. Ultimately, though, I don't think it's really necessary for the build. Your gear is going to be pretty sweet even without improvements, or with the improvements you can give it without perks. Crafting a Daedric bow would be badass, but you've already got the (faster) Bound Bow.
One/Two Handed - Sure, carry a sword if you want, but it shouldn't be necessary except in those cases where you're surrounded and your Destruction spells just won't cut it. Of course, in those cases, you're probably a little screwed anyway - so plan carefully! Bottom line, you're an archer, so aim to avoid melee.
Restoration - No question, there's tons of utility in this school. But an Arcane Archer typically doesn't need a lot of in-combat healing, since he's trying to stay out of melee range. Why cast a Ward when you can just, you know, dodge & shoot the bad guy? Turn Undead is probably the best contender for Restoration spells, and in draugr dungeons you're going to have a harder time getting distance from your enemies. It may be worth it to pick up a tome early on if you're worried about that sort of thing.
Alchemy - the archer's best friend. So why haven't I included it? Well, the Archery perks are pretty great on their own. With careful tactics and planning, you can get by with the poisons you'll find and a decent bow and arrows. That said, there are some devastating poisons - particularly for slowing enemies - so feel free to experiment if you don't mind working up and perking another skill.
Here's a sample build, up to the mid-40s in level: http://skyrimcalculator.com/#123879
And, since my own characters tend to peter out in the mid-20s, here's a build up to 25: http://skyrimcalculator.com/#123895
Shouts of note:
Become Ethereal - your panic button when an ambush was unavoidable. Retreat, heal, and come back for revenge.
Disarm - I haven't used this Shout, but I imagine it would be great for when that Orc with a battleaxe is bearing down on you...
Dismaying Shout - see my discussion of Illusion spells for why I feel this sort of thing can feel a little wonky, but the Shout requires less investment to keep it working, and you're pretty limited in how often you can use it in battle. I'd consider this, but there are other options that are more fun & thematic IMO.
Ice Form - a little cheesy, and I don't know how it really fits an Arcane Archer thematically, but you can think of this as a cheap paralyze before you get the spell.
Marked for Death - I like this; basically a ranged curse. Seems like more of an assassin/necromancer kind of thing, but you could also think of it like you're dispelling enemies' protections.
Throw Voice - a good distraction, but you may get similar results with Magelight. Nevertheless, this doesn't use up Magicka. Get creative!
Unrelenting Force - nice crowd control, but be sure to use it when you can make it count; the cooldown time for all three words can be a pain.
Whirlwind Sprint - when you absolutely, positively, need to get some distance for another few shots. This also has a few uses in dungeons.
Gear:
The single most important item in your inventory is your bow - or, rather, bows. Have several, and enchant them early, even if the enchantment is relatively weak. Merely recharging your weapon will help work up your Enchanting skill, which may be tough when you're running with the Warrior stone.
Be sure to carry bows of different weights, to take advantage of higher speed or higher damage. Sometimes you'll want a single, powerful shot; other times (when you are facing multiple enemies, and/or have Power Shot or other powerful enchants) you'll get more use out of a quicker draw weight. Hunting Bows are among the lightest available, and can be upgraded with readily-available items, even if you don't invest anything into Smithing.
The Bound Bow can serve you for some time, but at early levels it may be rather costly in terms of Magicka, and at high levels your own bows with specialized enchants will probably be more effective. It will find its niche once you perk it out via the Conjuration tree for effects like soul trap and banish/turn.
Arcane Archers can make great use of found items like potions and scrolls. Since my posted build doesn't use Alchemy, hoard those poisons, especially Paralyze. Potions for Health and Stamina are essential as well. Magicka potions will probably see less use, since you won't cast in combat often, but carry a few just in case. And of course, avail yourself of the various Resist potions.
As for scrolls, they can help fill in the gaps in your repertoire. Timed carefully, master-level Destruction scrolls can level a gang of foes who have managed to swarm you. Circle of Protection can help in those dungeons where you don't have room to kite. And the Illusion scrolls can allow you to focus on fewer enemies at once. As with every other aspect of combat, using scrolls is something an Arcane Archer needs to plan for carefully - switching from bow to scroll takes time, during which you don't want to be taking damage.
Staves are often handy, but less so for an archer, who can't fight and wield a staff at the same time. An exception here might be a staff of 'Wall of X'. With this, you can turn a hallway into a killing zone while you pelt away with arrows. Of course, timing is everything; you can't lay the groundwork for a killing zone when bad guys are shoving axes into your face. Walk carefully and plan ahead.
Quests:
I'm still contemplating this aspect of the character.
The Main Quest offers several benefits, though again, something about an Altmer archer/mage finishing the MQ seems a bit off. Grabbing Shouts and killing dragons is good fun, though, even if you don't advance all the way.
The Thieves' Guild grants access to an Archery trainer and some nice gear for an archer. A solid choice, especially if you plan to go the more stealthy route. For the build outlined here, however, and for the character concept I have in my own mind, it's not a great fit. The TG are jerks, the light armor they give you isn't going to help the build, and you can get nice bows elsewhere.
The Dark Brotherhood could be a good choice for similar reasons as the TG, but again it would be more for a stealthy character. Can't argue with the gear, though, especially the hood. If that's all that's motivating you to join, remember that there are other ways to get that gear...
The Companions could be good, but I have a hard time with the whole Nordic theme for my Altmer fighter/wizard/archer.
There are a host of minor and miscellaneous quests that would fit an adventurous gun for hire. I'm happy to hear if anyone has any specific suggestions, though be sure to use spoiler tags!
Followers:
I usually don't like followers. They get in my way and seem to glitch out at the worst moments. But if you want one, you can go ranged (spells/bow) or melee. Melee followers will get in your way often, which is terribly frustrating given the relatively slow draw speed of bows. Ranged followers are usually a better bet, but they can develop problems or quirks of their own.
Playstyle:
This is a kiting archer, who depends upon Archery perks and enchanted bows for punch. But that doesn't mean he shouldn't plan his approach. Especially in the close quarters of most dungeons, a good strategy is to cast a rune ahead into a new room, then wait for enemies to come and investigate while you cast your shield spell and nock an arrow. This allows you to avoid being surprised, and to somewhat choose your ground - being careful to note avenues for retreat/kiting, and any traps that can be used to your advantage.
If you can't take out your enemies with your arrows and you're running out of room, switch to spells, scrolls, or shouts to finish the job.
In open areas, a healthy Stamina pool will be needed if you plan to outrun enemies you just can't handle. But be sure to use terrain to your advantage as well, noting cliffs and rivers you can use to escape most foes.
If anyone has any suggestions or other thoughts, I'm happy to hear them!