Once Upon A Time... nvm not worth mentioning.

I have used this build in both ranked and non-ranked games. In non-ranked games, I will typically completely dominate games and own from start to finish. In ranked games, the competition goes up significantly, but your ability to dominate can remain just as high if played correctly. Also, this guide isn't just theory or talk, I use this build and have significant success with it. Here's some proof for ya.

Let's begin with a quick understanding of the role Ashe should play in League of Legends. Ashe is a carry champion, as a carry, this means that by late game, Ashe should be able to take out just about any champion in the game, with little difficulty, so long as she is properly protected. It is important that every team have a carry champion, and Ashe is a great choice for that role.

It has been my experience, that in order to fully maximize a champion's potential, you must minimize their weaknesses, and maximize their strengths. So let's consider many of Ashe's weaknesses, and see how we can counteract that.

1. Ashe has a poor mana pool, and regen.

Naturally, Ashe's mana pool is very small, and her regen is very low. Because of this, her abilities, that cost a pretty good amount of mana, are not normally spam-able (able to be used in quick succession). Greater Seals of Clarity are a great compensation for this, and allow you to avoid wasting gold on other mana regen sources.

2. Ashe is extremely squishy.

Ashe is a ranged champion, and this allows her to stay alive if played appropriately. However, you do need to be able to attack, or you might as well not play. A great way to compensate for this lack of survivability is to increase her crowd control power, and increase her run speed. This is why we select Boots of Swiftness and Greater Glyphs of Celerity. The Boots, allow you great movement speed to kite enemy champions, especially would-be assailants who think they can take a squishy Ashe down. The Greater Glyphs of Celerity, allow you to spam your crowd control ability Volley, and have a nice short cooldown on Enchanted Crystal Arrow.

3. Ashe is easy to counter, by stacking armor.

A common build for many Ashe players, is to go with lots of Crit. This is a terrible mistake, because if you ever face a good team, you will get absolutely eaten alive. Crit is unreliable, and without significant armor penetration, a high crit on a high armor opponent will top off around a pathetic 550(ish). While this seems like a lot, a good armor penetration ashe will get consistent 350-400 hits on that same opponent, and will crit upwards of 700-800. Black Cleaver and Stark's Fervor, are excellent choices for armor penetration, as well as armor penetration runes with Greater Quintessence of Desolation.

Debunking the Myths

There are a lot of terrible builds for Ashe. These builds are terrible for many different reasons, but lets consider them for a moment.

1. Crit Ashe. Infinity edge, Phantom Dancer, Crit Runes.

This version of Ashe has so many deficits it's not even funny. #1 You will lack too much armor penetration, and will find that tanks will absolutely destroy you. #2 You will run out of mana way too quickly reducing your survivability significantly. #3 You will not be able to spam your Volly ability, which makes you a one-trick auto-attack pony. By the same token, you will also have a very long cooldown on your ultimate, which is so powerful with a < 60second cooldown.

2. Lifesteal Ashe. Bloodthirster * 2-3

This build isn't terrible, so long as your runes are correctly chosen. But you will find that tanks will still be a sore spot. The reason I focus on tanks so much, is because, you won't have any problems killing other champions, whether they're melee, ranged carries, mages, it really doesn't matter. A high armor champion, usually a tank, is your biggest opponent. 50% lifesteal isn't going to help you enough if you're only hitting your opponent for 100-200 damage.

3. Survival Ashe. Merc Treads, Banshee's Veil, Dodge Runes w/ Phantom Dancer, HP Runes

The main issues with this build, is the lack of killing power. You will be successful with surviving a lot more. But Ashe has other methods of survival, while dishing out a lot of damage. This build will keep you alive, but this really isn't your role. Your role is to completely annihilate the opposing team when your tank initiates and protects you.

Okay, enough ripping on other builds, lets recognize what we need to do to be successful with building an Ashe. Here is the proper building order.

Item Explanations:

Doran's Blade
9 Attack Damage, 100 Health, 3 Life Steal. This is a pretty good starting item for Ashe. She is pretty squishy and so the extra 100hp will help in this arena. Additionally, 9AD and 3%lifesteal will help sustain you in a lane or mid. I used to get 2 of these, however due to the cost change, it is a less viable choice and I was leaning away from it for a while now anyway.
Boots of Swiftness
UNIQUE Passive: Enhanced Movement 3. These are your staple boots, you will be too slow with any other boot choice.
B.F. Sword
50 Attack Damage. This item is your first big purchase, and because Ashe kinda hits like a whimp early on, it is a great choice to really beef up your dps early game and mid game.
Zeal
20 Attack Speed, 10 Critical Strike Chance, 8 Movement Speed. There are a couple of reasons why I get this item so early in the build. First, Zeal gives a lot of benefits for the cost. At only 1195 gold, you get great attack benefits with crit and speed, but most significantly, you also get 8% additional movement speed. This will greatly increase your ability to kite opponents. Dive in and out of team fights as you get targeted. And avoid ganks that will definitely be coming your way.
Infinity Edge
75 Attack Damage, 20 Critical Strike Chance. UNIQUE Passive: Critical Hits now deal 250% instead of 200%. I bet some of you never thought you'd see the day that infinity edge was included in my Ashe build. Well... I didn't want to, I really didn't. However, it really a great item at this point in the game for Ashe. The total damage that you'll be ripping out with this item addition, especially to squishier champions, cannot be ignored.
Vampiric Scepter
12% Lifesteal. I don't always select this item at this point in the game. However, I will often pick it up here for a couple of reasons. The addition of 12% lifesteal to your already 3% gives you 15% which is a significant boost in lifesteal. Also, at this point you will have plenty of attack damage, and some added lifesteal will definitely help you in pushing lanes and teamfights where you're having to dive in and out because you're being focused when you show your face.
Phantom Dancer
55 Attack Speed, 30 Critical Strike Chance, 15 Movement Speed. Okay, so for Infinity edge to get its full effectiveness, you really need to have Phantom Dancers. Additionally, since you'll already be getting Zeal, it's not an overly expensive upgrade, and the added bonuses are significant.
The Black Cleaver
30 Attack Speed, 55 Attack Damage. Passive: Physical attacks reduce your target's Armor by 15 for 5 seconds, (maximum 3 stacks). This item changed significantly in the latest patch, and because of this, it has been moved in my build. The introduction of attack speed literally had me running around in my house saying "holy ****" to myself repeatedly. Then I realized the drop off of attack damage, the increased price, and the lower armor reduction and I realized it was a decently balanced item but still very valuable.
Stark's Fervor
20 Attack Speed. UNIQUE Aura: Gives nearby allies Champions 20% lifesteal, 20% Attack Speed, and 30 Health per 5 seconds regenerated. Reduces the Armor of nearby enemy champions by 20. Okay so here's the deal with Stark's Fervor. I don't necessarily think its the best item for Ashe, however I think it is pretty essential that you have a Stark's Fervor on your team. Since 9 times out of 10 I can't find an ally who will pick it, I usually end up with it. Stark's Fervor actually does fit into the build relatively well since it adds some armor penetration, and additional attack speed that you already have a considerable amount of. Also, its 20% lifesteal completely eliminates your need for Doran's Blade.
Banshee's Veil
375 Health, 375 Mana, 50 Magic Resist. UNIQUE Passive: Blocks one negative spell every 30 seconds. When you buy your Banshee's Veil, you're now full on items, so you need to sell one of your Doran's Blades. The key here is that you are replacing the hit-points that the Doran's Blades provided. If you don't replace those lost hit-points your survivability drops significantly. Banshee's veil is the best choice because it will help protect you from enemy crowd control effects, and AP carries who try to nuke you down quickly.

21/0/9 Is in my opinion the best build for Ashe. With 21/9/0 You'll lack enough mana regeneration, and with 9/0/21 you'll lack enough killing power.

Greater Mark of Desolation x9
15.03 Armor Penetration. It's important to have armor penetration runes. With full armor pen you gain an additional 25 armor penetration which is absolutely invaluable.
Greater Seal of Clarity x9
0.9 Mana Per 5 Sec per level (16.2 at 18). These seals will allow you to avoid wasting gold and item slots on any mana regen items.
Greater Glyph of Celerity x9
0.45 Cooldown Reduction per level (8.1 at 18). These Glyphs will allow you to spam Volley, and have a nice low approx 45 second cooldown on your Enchanted Crystal Arrow.
Greater Quintessence of Desolation x3
9.96 Armor Penetration. Armor penetration for the win.

Flash
Teleports your champion to target nearby location. Best survival spell for Ashe. This is your get out of jail free card if you make a poor positioning choice. It's also good for chasing if the situation calls for it.
Ghost
Your champion ignores unit collision and moves 32% faster for 14 seconds. This spell is great again for increased survivability, coming to allies aid quickly, and chasing running champions.

Principle #1

Use your Ultimate Often. With this build you will have a nice low cooldown on your ultimate. Use it often, and carve your way to kills. When I am 1v1 mid, I will ultimate a champion who has 100% hp to get them down to 30-50%. If they don't fall back and heal up, I will easily get the kill the next time my ultimate is up.

Principle #2

Volley is a spam skill. Yes you will have some mana issues, but you cannot let that stop you from spamming this skill. It is an extremely powerful way to hold groups of enemies at bay, protect allies who overextend, and farm minion kills.

Principle #3

You must get good at knowing when to toggle Frost Shot. Frost Shot is the difference between destroying enemy champions at will, and watching them run away laughing at you. You can't leave it on 100% or you'll always run out of mana, but you also can't forget to use it.

Principle #4

Hawkshot cannot go underutilized. This is a great ability for map awareness. I will often buy wards as well, but using Hawkshot often is very important. If you don't have anything you want to see when its not on cooldown, pick a popular jungle mob i.e. Dragon, Golem, Lizard, Baron, and check to see if anyone is there for a possible ganking opportunity. I was reminded by my friend Creevy not to take for granted that everyone would know this simple strategy: You Can Use Hawkshot To Scout Bushes.

Principle #5

Teamfights must be engaged carefully. I cannot stress just how important it is that you engage teamfights carefully. You are very squishy, and if you allow a couple of melee champions to dive you, you will die. Start team fights by staying at max range and spamming Volley. If an opponent overextends, wipe them out.

I hope that you enjoy learning to play Ashe in this way. Do not expect that you will grab this build and dominate immediately. There is a learning curve to this play-style, and you will have struggles. However, I am confident that you can learn to play Ashe effectively utilizing this build. Good luck! And remember to vote and comment :)