Hello! This is my 3rd build, and first detailed one. A lot of my information will be taken from www.LeagueofLegends.com so if any information is inaccurate, please, let me know. My other builds that I have to update are:

Karthas
Tryndamere

Things to know:
[MS] = Mystic shot
[RTB] = Return to base
[Ulti] = Ultimate
[BN] = Barron Nashor
[CC] = Crowd Control
[Juke] = Going one way and then suddenly switching to a different route confusing enemy chasers.
[Kite] = Focusing attention on you while avoiding death and hitting the enemy as well
[Blink] = If you played Warcraft III, you know this. Blink = flash or another kind of porting spell (different from Teleport). I am from DOTA as well as other MOBA or AOS type of games and I prefer to say blink, rather than flash.

Lore:

Ezreal was born with the gift of magic flowing through his veins. Ezreal, however, was also born with a much stronger sense of wanderlust. Put into school to become a skilled techmaturgist, Ezreal quickly became bored with magical studies. By the time the boy genius was eight years old, he had fully mapped out the underground tunnels of Piltover. The quality of his work was so great that the government of Piltover purchased his maps and salaried his services as Piltover's Grandmaster Explorer. This sealed the deal on Ezreal's path in life - he would eschew the arcane arts in favor of archaeology. Since then, countless of Ezreal's adventures have been written about as romanticized stories.

One of Ezreal's latest adventures, however, has brought him face-to-face with his other legacy - his latent magical power. While exploring the pyramids of Shurima Desert, Ezreal uncovered an Amulet of incredible power. Aside from the sheer size of the amulet (it was made for a being easily twice Ezreal's size), it allows the wielder to control and shape magical energy - provided a source of magic is in the vicinity. This allowed Ezreal to tap into his natural talent for magic without having to put any serious effort into it - a big win for the Prodigal Explorer. The drawback is that for some unknown reason, the amulet is attuned with summoning magic. Without warning, Ezreal may find himself acting as a champion for, as he puts it, ''a summoner hell-bent on resolving some irrelevant world-shattering League conflict.'' Still, Ezreal feels being summoned into a Field of Justice on occasion is a small price to pay.

''There's little time to study musty tomes when you're busy crawling around where the musty tomes originally came from.'' Such is Ezreal's credo.

Pros:

Very flexible character, fits the assassin role, damage per second and support
Very powerful the entire game
Buffs / Debuffs for Friend or Foe
Longest Skill shots in the game (Map wide ultimate, largest range for MS and Essence Flux)
Great Farmer (Last hitting is a breeze)
Early Harasser (+120 damage at level 1 for your MS)
Great Mid (Farmer, Early Ultimate to help side lanes)
Great Carry (Crazy damage, nuke down squishy in seconds)
Blink for sticky situations (juking)

Cons:

First to get targeted
Squishy
Slow base movement speed
Is harassed easily and pales in comparison to other mid carries such as Miss Fortune
If your not great with skill shots, this is not the champion for you

Skills

Mystic Shot - Ezreal fires a bolt of energy, if it strikes an enemy unit it reduces all of Ezreal's cooldowns by 1 second.
Ezreal fires a bolt of energy dealing 35/55/75/95/115 (+) physical damage (applies on-hit effects).

Ezreal's cooldowns are reduced by 1 second if Mystic Shot hits a target.

Cost
30/35/40/45/50 Mana
Range
10000
This is the skill that will eventually break your keyboard. The entire time during fights you will constantly mash this key. MS has one of the longest (if not the longest) ranged skill shots in the game. Meaning you can harass freely in mid while keeping a safe distance. This skill also costs a very little amount of mana, so you will be using this a lot.

Essence Flux - Ezreal fires a fluctuating wave of energy, damaging and reducing enemy champions attack speed and increasing allied champions attack speed.
Ezreal fires a wave of energy that damages all enemy champions it passes through for 80/130/180/230/280 (+0.6) magic damage. Affected ally and enemy champions have their attack speed respectively hasted or slowed by 20/25/30/35/40% for 5 seconds.

Cost
70/80/90/100/110 Mana
Range
10000

Not much to say here. If you are going AD like you should, this skill is worthless until late game. Better to just skip it for now. The attack speed buff/debuff is nice but honestly you don't need it till very late into the game.

Arcane Shift - Ezreal teleports to a target nearby location and fires a homing arrow which strikes the nearest enemy unit.
Ezreal teleports to a target nearby location and fires a homing arrow at the nearest enemy unit dealing 75/125/175/225/275 (+0.75) magic damage.

Cost
90/90/90/90/90 Mana
Range
10000

The long cool down will nearly kill you if you aren't careful when using this. This is my favorite utility spell since you can skip out on the flash for this. Use this to wall jump. The great thing is, is that it pushes you over walls that normally flash pushes you away. Using this as an opener for a 1v1 is risky because if they start killing you faster than you can kill them you are screwed, so use your head wisely and think before you use it. Another note is that people use this skill to rearrange themselves to use Trueshot Barrage, but I don't really think it is worth it. I rather just walk a little bit then use it, rather than waste a 10 second cooldown for a spell that will port me just a few steps away. There are times in which I do use it for repositioning myself, but more often than not I just walk if there is no hurry.

Trueshot Barrage - Ezreal channels for 1 second to fire a powerful barrage of energy missiles which do massive damage to each unit they pass through (deals 8% less damage to each unit it passes through).
Ezreal channels for 1 second to fire a barrage of missiles dealing 350/500/650 (+0.7) magic damage to each unit it passes through. However, it deals 8% less damage for each unit it hits (Minimum 30%).

Cost
150/150/150 Mana
Range
20000

Nothing much to say on this as well. I use this to start or finish a fight. Useful as to seeing when you are in top and say a fight breaks down on Nashor. If you time it right, you can snipe Baron and an enemy or 2 as well. I mostly use this to push.

Runes

9x Greater Mark of Desolation. You will eventually have armor pen with the Black Cleaver, but added with these, you WILL hit HARD. Tanks Beware!

9x Greater Seal of Clarity. I find that I run out of mana very quickly. With more mana, I am better able to last hit creeps, farm more gold and stay in lane longer which is equal to better carry potential.

9x Greater Glyph of Alacrity To be honest, you can pretty much fill glyphs for anything you want. You need more mana? Fill it up with Clarity runes. You want some critical damage? Throw in some Malice runes. Personally, I use attack speed simply because its the only complete glyph set i have (other than ability power). If you guys want to use any other runes, you are very much welcome to. From what I have seen, some people like to use cool down runes on this guy. Personally, I think they are a little redundant simply because you "q" knocks off a lot of the cool down and I don't really have a problem with it.

3x Greater Quintessence of [Frosty] Fortitude I use these instead of armor penetration to help you be less squishy early game and let you farm a bit longer. You can use armor penetration if you really want to, but this adds survivability where as armor pen adds damage. It honestly is the whole "Is tank Vladimir better or A.P. Vlad better?" Personally I like Tank Vlad because he can survive waaaaayyyy longer than an ap Vlad can.

Items

*NOTE*
After you buy your 1st Bloodthirster, you will have to (unless you like Wriggles / Doran's Blade) sell your first two items. The reason being is that I use Wriggles and Doran's to help early game lane domination. The wriggles would give you great life steal and the power to push mid tower very fast. The Doran's gives you HP as well as early damage and life steal.

Bloodthirster: +60 Attack Damage +15% Lifesteal Passive: Gain an additional 1 Physical Damage and 0.25% Lifesteal per kill. Maximum of +40 Damage and 10% Lifesteal. Bonuses are lost upon death.

What can I say? 100 attack damage, 25% life steal of an already 400-500 damage MS. The reason I go for this instead of Infinity Edge is simply because this gives more benefits than I.E. I just don't find my self hitting critical hits as often as to find myself buying an I.E.

Madred's Bloodrazor: +30 Attack Damage +40% Attack Speed +25 Armor UNIQUE Passive: On hit, deals Magic Damage equal to 4% of the target's maximum Health.

This is what made AD Ezreal so overpowered Pre-Sona-Patch. I was literally tearing through (pre-nerfed) Mordekaiser like he was a piece of paper. The early proc on Madred's Razor made last hitting creeps a breeze and helped me farm more gold. The Bloodrazor however, adds a small amount of attack, decent attack speed and armor which are great, but the real benefit is the 4% of maximum health. You will be ripping through tanks like they were nothing if they didn't build properly against you, and even if they did, you would be doing so much damage it wouldn't even matter. Those squishies that stack health such as an AP Vlad will fall beneath your feet,

Black Cleaver: +55 Attack Damage +30% Attack Speed UNIQUE Passive: Physical attacks reduce your target's Armor by 15 for 5 seconds (maximum 3 stacks).

Great damage and the cheapest B.F. Sword build. The attack speed you will have from this is a great help and should boost the effectiveness of Ezreal a lot, even without mentioning it shreds armor.

ALTERNATIVE: Mercury Treads: 25 Magic Resist. UNIQUE Passive: Enhanced Movement 2 and reduces the duration of stuns, slows, taunts, fears, snares, sleeps, and immobilizes by 35%.

If there are a couple of caster champions on their team that have cc, I suggest picking these up. Especially if they have a Galio on their team. With these babies, you can literally walk or blink out of his ultimate. It's pretty handy.

Reason why I do NOT go for Trinity Force-
Trinity force does not take the added item buffs such as Infinity Edge and add the 150% with it, meaning it only adds your BASE damage. In Ezreal's case, it is 103 at level 18. At level 18, you would only be doing a bonus of around 154.5 magic damage, which you have no penetration for and it only procs once every few seconds. Sure if a team is filled with squish champions, you can go for it, but you will not get the same damage output than if you used say a Bloodthirster that adds 100 physical attack damage to your MS as well as your Auto Attack. Trinity force procs only once in a couple seconds, while Bloodthirster and Madred adds constant damage. An item that costs about 4000 to get around 150 magic damage boost, I'd much rather go for a Bloodthirster that costs 800 less gold, constant damage, and 25% life steal. Sure the proc for the slow on Trinity Force is cool, but I'd rather just pick up red buff than sacrifice an item slot for a slow. Also, when they took out the reservoir system with the Trinity Force,

This is solely MY OPINION. I used to use Trinityforce, but I yielded better results from having an extra Bloodthirster IMO.

Lane Phase
I highly recommend going mid with this guide. You are the ranged AD Carry for your team and you need to be farmed, more so than anyone else. (Maybe besides Ashe, however I think she is just kinda dead weight unless the player is farmed well, or is incredible with her arrow.)
Beginning:

Grab a Doran's Blade for the extra life steal, damage and health as well as a health pot and put a point into MS then head towards mid. Use your auto to damage creeps and use MS to last hit another creep.
Basically, you hit one creep with your auto and last hit a different creep with your mystic shot while your creeps beat down the one you are going to mystic shot. An ideal target for your auto attack would be the ranged creep and use MS on a melee creep that targeted by your own creeps. Before you return to base, farm around 1000-2000 gold so you can at least buy a Madred, if not a Wriggle's. After you buy your wriggles, you will pretty much never need to base again unless you are out numbered in mid by 3 to 1, and even then, they should not even be able to touch you. When you do go back, be sure to have farmed enough for your Berserker's Greaves. You should also buy a Recurve bow for the added attack speed which you need since you will be getting damage first, before you move into attack speed. Before returning to base (assuming lvl 6) fire your ultimate down your lane making sure to hit the creeps. Don't worry if you don't hit the enemy hero, this is meant to push your creeps a little further and give you time so you can head back to your lane. If you time it right, you can hit the creeps that just finish spawning at the enemy nexus and when they get to fight your creeps, they will be at half health.

Mid Game:
You should of already have a kill or two (usually 5). Stick with your team!!!!! I cannot stress enough how important it is for your team to be grouped up and move around the map, even if you ward, take at least one person with you. If the other team finds one person alone, If I were them, I would go after that one person and kill them, then focus on a 4 v 5 team fight or rush a tower. By this time, you should of already built your Black Cleaver and Madred's Blood Razor. If you also have a vampire scepter, you can solo Baron. If you do decide to solo Baron, be sure it is clear of any wards or mushrooms or traps etc. Baron buff can Make or Break an entire game. I've seen and been the team that was completely dominated by the other team and won because we aced them with the Baron buff. BARON BUFF IS NOT AN AUTO WIN. BE CAREFUL OF WHEN YOU HAVE IT, IF YOU ARE LOSING AND ARE UNDER FARMED, BARON IS AN EQUALIZER, NOT AN ADVANTAGE!!!

Late Game:

You should be close to or finished your build already. This is around 40 minutes into the game. If you haven't done the Baron buff yet, get it ASAP when it's safe to do so. ALWAYS MOVE WITH YOUR TEAM, NEVER GO ANYWHERE ALONE!!! I found an excellent late game guide by 5HITCOMBO on the LoL Forums.
http://www.leagueoflegends.com/board/showthread.php?t=470669