I got into playing Ezreal for two reasons: I love skillshots and he looks freaking awesome. Also, he does the Hare Hare Yuki, but that's a whole other story. He is basically a ranged acrobat, harassing from a distance, dodging, then moving in for the kill. Ezreal can be played in pretty much any way and is a jack of all trades. However, he is possibly the single most item dependent champion in the game. A badly built Ezreal will get curbstomped into oblivion.

PROS:
-Excellent Harasser
-Low cooldowns
-Is untouchable when played right
-Can be played either AP or AD
-Cross map Ulti
-Very fun and rewarding to play
-Last hits like a mofo
-Can mid
-Only skillshots

CONS:
-Pretty squishy (unless you build right)
-No crowd control (unless you build right)
-Weak early game
-So item dependent, it isn't even funny.
-Only skillshots
-Hard to jungle
-Requires much practice

This particular item build has 4 stages: Start, Hybrid LV1, Hybrid LV2, and KILL ALL THE THING! You can grab the items of each level in any order depending on the situation.

START:
Boots of Speed- You need to be able to get in and out ASAP early game. Early game Ezreal can get destroyed unless you can outrun the enemy. I always bring some health potions to keep myself in lane early game. It's too easy for the enemy to take advantage of your absence to farm your minions.

Ionion Boots are next. I like them because Ezreal is ridiculously skill dependent and his passive makes rapid fire abilities all the more important.

HYBRID LV1
Zeal- I get it early to last hit minions more easily, be able to run just a bit faster, and a little extra crit chance never hurts.

Sheen- Absolutely vital. Gives Mystic Shot and Arcane Shift a huge boost in power and, personally, the glowing red gloves look awesome.

Phage- An unusual item on Ezreal, the extra health improves survivability and the CC effect can save your ***. Don't rely on it though, it only has a one in four chance of working.

HYBRID LV2
Phantom Dancer- At this point in your build, you will begin to destroy. I get Phantom Dancer first for the same reason I get Zeal first: more last hits= more gold= more items= KILL ALL THE THING!

Lich Bane- Boosts AP, Mana, and Movement speed. The passive is what we want it for, and your Mystic Shot and Arcane Shift will be all the more powerful.

Frozen Mallet- Massive health boost and a guaranteed slow effect. This is a godsend when being chased. Just fire one shot and keep running.

KILL ALL THE THING!

At this point in your build, you'll be hitting like a runaway truck. I have only lost one game once I'd gotten to this point in my build.

Hextech Gunblade- AP, AD, lifesteal, and spellvamp. With the massive damage you'll be dishing out, you can enter a team fight and come out with more health than you started with.

Rabadon's Deathcap- INSANE AP! With this, your spell vamp will fill up your health bar and your Lich Bane will make the enemy team flee in terror.

Rising Spell Force- This is your passive and your greatest asset if your jungle or farm. Spam your abilities and your attack speed will go up with it. If you fire your Ulti at enemies at the start of a battle, you will start the battle with the advantage. Also, "Rising Spell Force" is an awesome band name.

Mystic Shot- This will be your most used ability. It had decent damage and even scales off AD and a little AP. However, it's most powerful attributes are its range, cooldown reduction, and Cooldown. You can harass from a distance and with Sheen/ Lich Bane it can wreak the enemy team. You can fire it ~3 seconds and use it to boost your Rising Spell Force. Every time you hit, you reduce your other skills' cooldowns by one second. THat might not sound like much, but you can potentially reduce all other cooldowns by 25% by spamming Mystic Shot.

Essence Flux- An often overlooked ability that can harass and support. It will boost your teammates' attack speed and lower your enemies'. It does good damage, especially at higher levels. However, it has one trait that makes it stand strongly alongside Ezreal's other abilities: it only hit Champions. Many enemy champions will hide behind a line of minions and will be protected from your Mystic Shot. They will often get closer than they would otherwise and will be ripe for a quick shot to make them back off.

Arcane Shift- This ability has three functions: a way to do additional damage, to get in a better position to fire off your other abilities, and is your GTFO button. It does only a little damage and is primarily just used for the teleport. You can use it to chase down a champion getting away with a little health OR you can use it to quickly get yourself out of bad situations. I cannot count the times I've survived only because of this ability.

Trueshot Barrage- An amazing Ulti. Can be used in three ways: To damage an enemy team before a teamfight, to destroy a massive wave of minions, or to nail an escaping enemy champion. I prefer to use the last one, hitting recalling champions and dashing the hopes of escapees. It scales off of both AP and AD, so it will be powerful either way. It takes some practice to hit crossmap, bu there are few things more satisfying than to hit a champion 5 feet from their fountain.

COMBOS
My favorite technique is: Arcane Shift into optimal position, Essence Flux to do damage, Mystic Shot to reduce all cooldowns, and finally Trueshot Barrage to finish off the enemy. Late game, you will take off at least half the enemy's health bar. If you are iffy on whether or not you can survive, save the Arcane Shift for escaping.