Ezreal is falling back nowadays in competitive games. I still think he is a good pick in Solo que or normal games. With this build you can have surprisingly nice burst damage from the early levels, if you land your abilities the right way.

ps: It's my first guide, don't be too rude on me. :)

The normal AD Carry runes and masteries fit him very well, nothing special or new here.

Trinity Force: Its a bit expensive as a must buy item but gives you nice overall damage.
Phage or Sheen first? If you doing good, and want to bully your lane, buy sheen first. If your are even, or falling behind, buy phage so u can stay longer in lane.
Essence Reaver: It's a fency new item which helps Ez a lot. Lifesteal is always good to have on an ADC, and the passive restores your mana, allowing to do more combo's.
Last whisper: Against tanky teams a must buy item.
BoTRK: Gives you good sustain and attack speed. Most useful in dueling, and objective killing.
Infinity edge: Gives nice dmg, and critical chance. I dont think it's a core item for Ezreal, of course can be a good choice if you so far in lead that can't spend your gold.
Ghostblade: If you choose to buy IE, buy this one too, more critical, more synergy. Also gives you armor penetration.
Defensive items: Of course it's situational, against Ap heavy teams buy Banshee's, if u just wanna have some overall safety buy Guardian angel.

Passive: Rising Spell Force
Hitting a target with any of Ezreal's abilities increases his Attack Speed by 10% for 5 seconds (effect stacks up to 5 times).

Common thing to think passives are not that important. Of course they are! For example: Your opponents in lane are going back to base. You want to deal some damage on their turret. Stack up the passive with hitting minions with q or ultimate. They going to be surprised how fast they lost their turret.

Q: Mystic Shot
Ezreal fires a bolt of energy, dealing 35/55/75/95/115 (+1*AD) (+0.4*AP) physical damage (applies on-hit effects). Ezreal's cooldowns are reduced by 1 second if Mystic Shot hits a target.
Cost: 28/31/34/37/40 Mana
Range: 1150
Cooldown: 6/5.5/5/4.5/4 Seconds

You should max this ability first, and practice to land it as many times as it's possible. Awesome poking (one of the bests in the game). Low mana costs let you to use it for lasthitting minions, if u don't wanna get closer to the enemy. Don't forget the 1 sec cooldown reduction every time you hit an enemy unit (it can be very handful chasing down enemies).

W: Essence Flux

Ezreal fires a wave of energy that damages all enemy champions it passes through for 70/115/160/205/250 (+0.8*AP) magic damage. If Ezreal or his Allied champions are hit by the wave, their Attack Speed is increased by 20/25/30/35/40% for 5 seconds.
Cost: 50/60/70/80/90 Mana
Range: 1000
Cooldown: 9/9/9/9/9 Seconds

In early game this ability is not that important, but in teamfights can mean a big difference if you use it well. Just try and hit as many members in your team with it as you can. In laning you can have an attack speed buff for yourself, if u casting w and then immidiately e to get in front of the wave of essence flux.

E: Arcane Shift

Ezreal teleports to a target nearby location and fires a homing bolt at the nearest enemy unit, dealing 75/125/175/225/275 (+0.75*AP) magic damage.
Cost: 90 Mana
Range: 475
Cooldown: 19/17/15/13/11 Seconds

Because of the long cooldown you need to use it in the right moments. There is 3 way to use it:

Catching your w: as i wrote it before, you can use Arcane Shift to give yourself an attack speed buff.

Attack/Forward: Make sure that you have vision before you jump in somewhere, cause as an ADC you are squshy as a Chihuahua. If you have nice vision always jump next to the target you want to hit ( the bolt hits the closest enemy unit as well as Champions or minions.)

Defense/Backward: If you pushing out your lane it's alwals good to have Arcane shift avaible. If you get ganked, with a well timed teleporting they just going to waste their time and gain nothing for it. The range is not too big (475) but you can go trough every walls where you can flash through.

R: Trueshot Barrage

Ezreal channels for 1 second to fire a barrage of missiles dealing 350/500/650 (+0.9*AD) (+0.9*AP) magic damage to each unit it passes through. However, it deals 10% less damage for each unit it hits (minimum 30%).
Cost: 100 Mana
Range:Global
Cooldown: 80/80/80 Seconds

The Banana can change almost everything on the map. Since it's global with a well placed ultimate, you can help out your top or mid laner if they get in trouble. Pick up kills when the opponents think they are safe under their turret and try to recall with low hp. In teamfight try to hit the whole enemy team, prioritazing their carries. Steal a baron or a dragon with a well timed hit. You can do really cheeky stuff with that ultimate.

Pros:
High mobility
Good burst after a couple of items
Global ultimate
Nice AoE dmg
Insane poking
Passive gives attack speed ( no need for AS item)

Cons:
Skillshots (if you can't land your q's you are useless)
Squishy (as all adc)
No CC (you rely on your support in this)