Not Updated For Current Season
Having played Ezreal a fair bit I've tried many different builds. I used a AP build pre-patch, which obviously isn't very viable anymore.
I used an AD which turned out to be very similar to one I found on here (great minds think alike?).
However this one I find the most success, not as a carry but as a team support-carry. This build is more towards high DPS in team fights and ganking.
If possible grab the solo lane, but this build doesn't depend on creep feeding so don't feel the need to argue for middle, it's helpful not necessary. Having played mostly solo or with a premade of 1-3, I've quickly found out to ignore all ignorance and do not argue with your teammates. Just stop talking, they can't argue with no one.
Early game laneing is heavily dependent on AD, meaning your skill leveling is going to look a lot like an AD build. Q > E > Q > E > Q > R, making sure to harass as much as possible. If you're mid lane, I try to wait for my ultimate (don't be afraid to use f1-f5 for Trueshot Burrage shots across map, you'd be surprised how many people feel safe on their turret at low HP) once your opponent has left middle lane, I try to grab gank bot or top, usually choosing whoever needs for help, or just push the lane and give them some help.
Assuming your not laneing against Herm you'll have a bit before they push your tower. Pushing 3 bot or top will also force the mid lane to come repel the push.
If you successfully gank someone you should have enough money to grab boots and start on Sheen. Start leveling your W accordingly, grabbing it every 2 skills interchanging between Q and E.
Now a few items are dependent on your playstyle and their comp. First of all a Doran's blade is completely dependent on me relying on my Q early game. You can easily grab a Doran's shield, or if you don't grab Clarity, a Doran's Ring.
Your boots are going to depend on their team comp and items. If they happen to have a lot of slows, taunts, stuns, etc. I like to grab Mercury Treads. However you can easily abuse your WER if they don't magic resist using sorceror's boots.
Above all else the most necessary is just the boots for mobility. Don't feel the need to upgrade your boots right away if you're low on gold.
After I've finished Guinsoo's rageblade it's merely a preference. I'll grab Hextech Gunblade if I'm being fed and can support the item, or I'll grab Nashor's Tooth if I'm a little less farmed/fed.
I'll finish it off with a Frozen Mallet for durability and slows.
Your team is VERY dependent on your DPS in a team fight. Make sure your E is available at all time. I like to run a QW during the engagement, and E backwards out of the fight if at any time you're threatened.
I'll toss my Ult out if they are aligned in a manner which will greatly increase it's DPS or else I'll continue QW, praying on the squishiest with the highest DPS.
You have a quick cooldown on your ultimate, don't be afraid to use it.
Of course with my build I do not grab a Mana Regen item, so I'll either heavily depend on Blue Buff (which may be a challenge with a jungler in the game) or I'll grab clarity.
Exhaust is a must for me, it greatly helps ganking, which is a good way to feed Ezreal early game.
I wouldn't suggest Ghost or Flash, as Ezreal already has ways to opt out of a gank rather easily.
Ignite can help for Ganks but to be honest building Ezreal this way shows up largely in team fights and Ignite becomes rather useless then.
Clairvoyence is a under appreciated Summoner spell. If you do happen to grab it use it accordingly:
If a jungler is in the game, spam your Clairvoyence on the enemies Red and Blue buff early game, and on any buff in the game late. If not you can use it to protect yourself or teammates from a gank, and once in a blue moon you can catch someone whos ran away recalling and hit them with your ult for a free kill. Remember they can see the eye so try and use it behind a bush where it becomes hard for them to see it. The range on Clairvoyence is quite large. For example when I tell you to spam it on Blue and Red buff, I would suggest doing it straight across the bush from the Gollem. You'll still see the buff but he will not see the Clairvoyence.
You can justify any Summoner spell, but it's merely a preference.
My quintessence are as followed.
The later is a hunch. I used to build Ezreal as a Crit AD, which worked out fairly well, but he still occasionally crit's hard using this Hybrid Build. Nothing is set in stone.
Red: I'll usually use Cooldown reductions and Health.
Blue: I'll grab Crit and Attack Damage.
Yellow: I'll grab health regen and Dodge chance.
I'm going to drop a quick reminder here.
Remember that Youmuu's Blade and Hextech Gunblade both have unique Actives, so use them as often as possible.
Hope this helps,