Wow...prepare for a deluge of suggestions, most of which I suspect will read along the lines of "it's nearly impossible to play *just* an archer, as there's only one bow weapon line and a few bow-ish related class skills, meaning that you'll max out your bow line options very quickly and end up with about 100 sp's to spend in other things, plus you'll want to be more versatile than just using a bow all the time blah blah blah..."

sadly, a little of the above is basically true - your skills in ESO are divided into 3 lines of class skills, currently...*counts on fingers for a moment* 6 weapon skill lines (2h, dual wield, weapon and shield, bow, destruction magic, heal magic), and a veritable cornucopia of guild skills, racial lines, crafting lines, etc, etc...

there's only 1 bow/archer ONLY weapon skill line, it has 5 skills, 1 ultimate, and I think 4 passives in it - using up a grand total of maybe 24 skill points, which you should have by maybe level 15, possibly 20 if you skip all the skyshards somehow

the point being, you can definitely be an archer, but it's going to use up only a fraction of your total character abilities, and for utility reasons as well as just the fact that there's only so much you can spend on bow anyway, you'll probably want to expand your idea for an archer to include some other skill lines

all of that having been said, I think nightblade might be your best bet for a bow character - a couple of the class skills in the NB line are kind of archer-like, and the fighter's guild skill line has some crossbow skills, and possibly the pvp line or the undaunted line might have one or two each, I can't really remember

I don't recall for certain, but I don't think sorc or templar have anything specifically bow or x-bow like - I still haven't played my dragonknight so I'm less sure on that one

that's the best advice I can give off the top of my head - hope some of it helped

I've main a Nightblade bow build before it was cool ( it is still not cool but hey, who cares! ) and for what I know, Nightblade does have the best complementary play style for a bow build.

however, the bow requires you to either have a secondary weapon that buffs you, which in this case, either the Two hander or Dual wield. if you do plan to use your bow as your primary weapon though, the two hander has better buffs that you can handle as a bow user. it's basically a stat weapon.

an example of a bow / 2hander build would be:

Poison Injection ( opener and execute )
Venom Spray (AOE)
Relentless Focus ( sustain and additional dmg)
Camouflaged Hunter ( crit buff while slotted )
Shadowy Disguise ( initiation with guaranteed critical hit )
Toxic Barrage ( Bow ultimate )

Rally ( brutality buff and HOTs )
Double Take ( minor damage mitigation, speed buff )
Reaper's Mark ( heal and berserk buff on kill, debuffs enemy )
Rearming Trap ( critical damage buff )
Utility slot - Shadow Image / Mass Hysteria / Crippling Grasp / Siphoning Attacks
Aggressive Warhorse / Shooting Star / Soul Harvest / Flawless Dawnbreaker

depending on the situation, you can slot in Mass hysteria or shadow image for escape. a well timed teleport back to the image might save you, and it gives a minor maim debuff on the enemy too. Crippling grasp are really just for chasing down a prey but do bear in mind that both crippling and double take won't stack, unless you decide to go Mirage for more mitigation. Siphoning attacks are really for sustain and it pairs well with Relentless focus.

ultimate wise, it's really up to you. AW for more group play, shooting star for that finisher and ultimate generation so you can proc more Toxic Barrage. don't think Dawn breaker will work well here but it's an option. the other Nightblade ultimates are not as useful to you as a primary bow user but its all up to you. I've had some success with Soul Tether since it does an AOE stun that is quite useful in controlling groups as well as having good heals.

Hope this helps! - RavenSworn.