Ill try to keep my changes from the original build mentioned here before hand in order to give people who have seen the build a quick sense of what changed.

-- The recommended Hextech Gunblade has been changed to a Bloodthirster. As mentioned in the guide, this final item is really up to you, but some people have a made a good argument for getting a Bloodthirster.

This build is for those who like to play Ezreal by using his spells more than just auto attacking. The cooldown on Mystic shot becomes reduced enough so that it is easy to max out the 5 stack of Ezreal's passive, giving him the ability to also auto attack quickly.

The item order is simple. First I work my way to a tear of goddess. This allows you to spam your Mystic Shot as much as possible without going OOM instantly. Afterwards, I go for the basic boots to allow surviability. Run speed 1 + Arcane Shift+ flash should allow you to survive almost any gank attempts made against you.

Next I go for a quick manamune, which by that point increases your attack damage from around 80 to around 120. This significantly increases the damage of Mystic shot.

After the Manamune, aim to get your ironia boots. Some like to use Beserkers Greaves, but personally, Ezreal's auto attack damage is not great enough. I prefer the 15% cooldown reduction which allows for quicker Mystic Shots which in turn allows for quicker Arcane Shifts.

The next item is a sheen. Although this build mostly revolves around Mystic Shot damage, the sheen increases your damage Significantly when trying to go in for a quick burst.

After the Sheen, try to get a B.F. Sword which will build into a Black cleaver. Ezreal does not hit as hard as many other carry characters, but with the armor penetration and attack speed increase from the black cleaver, even tanks will go down relatively quickly if Ezreal can Kite or chase them. Other builds seem to have higher armor pen, but keep in mind that that you get another 45 armor pen after 3 shots when you have a black cleaver. This means that anyone who is not actively stacking armor at level 18 will be brought down to an effective 0 armor against you.

Next, is getting a phage, the phage is important because a) it slows down the enemy, and b) along with the sheen, it builds into the trinity force.

For ezreal, the trinity force is a must. Granted not all games will last long enough to allow you time to purchase the item, but If you manage to get it, your survivability and damage output will increase.

After the trinity force, the item choice is mostly up to you. If you can survive quite easily, then I suggest just getting more attack damage items. I tend to go with a Banshee's veil to avoid getting stunned or hit really hard by the enemies' ultimates.

The final item can be a Bloodthirster. The additional attack damage combined with life leach can be an effective tool. It's simple to build and has good rewards for the gold spent. This last item is up to you.

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EDIT

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Hextech has been removed from the build but you can read it here

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The hextech gunblade is also up to you. A game rarely lasts long enough to get this item, but if it does, the spellvamp, lifesteal, and additional AD+AP is always nice.

The skill sequence is simple. Mystic shot, then Arcane Shift, then Essence flux. Some dislike essence flux in the beginning because of its high mana cost and because it does not scale with AD. I have found it useful for quick ganks at early levels.

After the first 3 levels, get Mystic shot and your Ultimate whenever available. If Mystic Shot is maxed out or unavailable, get arcane shift. Once Mystic shot is maxed, if you cannot get a point in your Ultimate or in Arcane Shift, put points in essence flux. At late game it does enough damage and also helps to survive those other carries that stack life leech and attack speed.

The runes are also fairly simple. Get as much armor pen as possible. This is KEY for early game killing and also to kill those pesky tanks. I like to get cooldown reduction to Mystic Shot quickly.

Mana regen can be replaced by whatever you want. I have played without any mana regen and dont really have a mana problem past level 7. The regen is mostly to keep you going strong for the early levels.

Pros: You can easily kill any enemy (Usually). The exception is those high health tanks. They are not this builds strong suite but they are still killable. You just have to be careful and kite them.

The build provides excellent range damage. If the game lasts till you are level 18 and have at least up to your trinity force, everything (except HP tanks) will feel squishy to you.

Cons: Those HP tanks. Try not to face them alone...

Also, since this build relies heavily on Mystic Shot damage, you NEED to be good with your skill shots. If you cant land your skill shots, your auto shots don't do enough damage by themselves to kill most enemy champions.

This having been said the build is harder to play than some other Ezreal builds, but with some practice it can be very strong.

At early levels, like many heroes, try to just aim for that last shot. Luckily, since you get your tear of goddess early on, don't be shy about using Mystic Shot freely. Your mana should be quite good, and every time you use a spell, it will increase by 4 (up to a 1000).

I try to maintain a 5 stack of ezreal's passive in order to auto attack as fast as possible. Farming gold usually isn't a problem. Try to froce enemy champions to keep their distance by taking shots at them with mystic shot and essence flux. If you get them to about half health, you can usually arcane shift in for the kill.

Jungling is not recommended with this build. Your health pool is too low at early levels. Once you hit level 10 or so, you can go into the jungle and kill monsters, but it usually isnt necessary.

Don't be afraid to spear head and attack. If the enemy turns on you, just arcane shift to leave. If you have a team mate that can stun/trap the enemy, then a kill is usually guaranteed.

Try to keep your distance and let your team mates be in front of you. Your health pool is not large enough to take damage from more than one enemy champion.

This is mostly up to you. I recommend Exhaust and Flash. Both of these spells can be used offensively as well as defensively. Some people like ghost over flash. Others like Heal over Exhaust. The heal can be very useful early game, but I tend to play aggressive and if I can keep my enemy from running away then I tend to prefer that strategy.