First off, before I get into the details, we must first understand Ezreal as a champion and why we would choose to pick him.
In comparison to other AD carries, Ezreal serves as an all-around versatile ADC that excels in all areas. He has a strong early game because he can easily out-trade other ADCs, harass well, poke / farm from a distance, and escape easily. This strong early game allows Ezreal to mesh well with most other supports. This then transitions to a stronger mid game, where he is able to poke very easily, out-duel most if not all ADCs, and kite extremely well in fights.

Ezreal focuses on more of his abilities than flat-out autoattacks for his damage output. His Q is his primary tool for damage alongside with his autoattacks. Essentially, his Q is used for kiting the enemy (along with damage), and his autoattacks are used to follow-up with damage. This, however, does gimp his damage late game, because he will have to use his abilities to come into full effect, which limits potential crits, while other carries can simply autoattack and crit.

Masteries
The 21 points in offense provide the damage output a carry needs. The crit masteries Lethality and Frenzy are almost essential to most carries because of the fact that all ADCs eventually focus on crits. Alternatively, if choosing to take Ignite instead of Barrier, take a point into Summoner's Wrath and remove a point from Summoner's Resolve
The 9 points can be placed into utility, defense, or both. If opting for utility your ability to go all-in will be gimped but your harass and lane sustainability will be much stronger. If opting for defense you will have more problems with mana but your all-in will be much stronger.

Runes
9 AD Marks and 6 ArPen 3 AD are both good on Ezreal. 9 AD marks allow for strong last-hitting and lane control. 6 ArPen 3 AD allow for stronger damage potential on champions but weaker farming. ArPen is weak on minions because they have extremely low armor values which pose ArPen as unnecessary. 9 Armor seals are core on all ADCs because they mitigate damage from enemy ADCs, enemy supports (autoattacks), minions, and junglers. No exceptions. 9 MagRes glyphs and 5 MagRes 4 Mana Gen/ lvl are both good as well. the 9 MagRes mitigates damage from enemy carries and mostly damage from supports. If you're against a relatively weaker magic damage harassing support such as Nunu & Willump, you can also take Mana Gen/ lvl to help with mana problems later on in the game. 3 AD quints are quite good for controlling your lane and last hit under tower. Alternatively, while starting Doran's Blade starting with three or two Lifesteal quints helps greatly for lane pressure as the lifesteal makes up for the missing sustain from Doran's Blade.

Summoner Spells
Always choose to take Flash and another spell. Flash is essential for almost all champions, and even though Ezreal has a built-in blink, Flash is still best on him because of the double-blink potential ability to go all-in in more dangerous situations. For the second summoner spell, Barrier, Cleanse, and Ignite are all superb. I prefer Barrier because it provides a shield that cannot be mitigated by Ignite whilst Heal can. Cleanse is also strong against strong CC enemies and allows for quick re-positioning. Ignite allows for stronger all-ins and gives larger killing potential.

Sequence
Take your first ability AFTER you go to lane, not just at your spawn pool. If you're in a sticky situation and get caught before minions spawn quickly add Arcane Shift to escape without having to burn Flash. Elsewise take Mystic Shot at level one because it's most effective for helping a teammate deal damage to red buff or counter-harassing an enemy if they're stupid enough to engage at level one. After that take Arcane Shift at level two for the escape and gap-closer and another point in Mystic Shot at level three because Essence Flux accomplishes little early game. Always add Trueshot Barrage whenever you can because its your ultimate. Aside from that get Mystic Shot whenever you can because it's Ezreal's most effective damage output from early to mid game. After that Arcane Shift becomes the priority because the lower cooldowns help during skirmishes, engages, and teamfights. Finally max Essence Flux last because the damage is minimal and the attack speed buff is seldom necessary. Of course, when playing in competitive if you're going for a heavy push comp that isn't as reliant on damage output / skirmishes maxing Essence Flux would be better over Arcane Shift.

Starting with Doran's Blade is strong because of the strong sustaining capability with the lifesteal quints + Doran's Blade lifesteal and strong engage / harass with the AD bonus. Earlier on aim if necessary rush Vampiric Scepter and Boots because they provide you with more sustain for lane.
Aim to purchase Bloodthirster, Infinity Edge, or Blade of the Ruined King as your first big item depending on what kind of build you're aiming to run. Get Statikk Shiv for farming and pushing ( Zeal vs Avarice read misconception section) or Phantom Dancer (read Shiv vs PD misconception section) if you're going for the mid game crits, Trinity Force for utility if you went for Blade of the Ruined King first, or Blade of the Ruined King if you went for Infinity Edge first. Last Whisper will always be necessary sometime after the 3rd purchase because that's when enemies start to stack armor. Note that even if enemies don't have any armor items in their inventory they might buy it any second, so aiming for a Last Whisper will always be good. Generally, if wanting to purchase a defensive item, it will be best near fifth or sixth purchase. Each has their own purposes. Guardian Angel and Warmog's Armor are great against burst because of the resistances + passive and the effective health, respectively. However do note that GA will be useless if the fight ends before you respawn. Mercurial Scimitar provides a decent amount of damage along with an instant Cleanse which helps greatly for re-positioning and escaping. Banshee's Veil is chosen when an enemy team has one game-breaking spell that will kill you for sure, such as Malphite's ultimate or Blitzcrank's grab and also helps for general bulkiness. Frozen Mallet is a more offensive defensive item that grants you a decent amount of health and a great slow which helps for kiting.

Lane Phase
Ezreal has an extremely strong early game. Earlier on Mystic Shot and an auto-attack gives really strong harass. Ezreal is extremely strong at dueling so most enemies will choose to back out of engages. Keep in mind to not go too far as enemies can easily counter-harass you if you go too far. Generally a good marker is where the minions are at. In most situations you will always aim to stay behind large waves of minions when harassing or going all-in.
If pressured heavily (from jungler or ADC/ support) you can also choose to play defensively, last hitting with your Mystic Shot from a distance. Once you hit level 6 your all-in potential increases by a great deal. Trueshot Barrage is a great damage tool for cleaning up enemies or initiating a skirmish.

Lane Control
Lane control is very important as an ADC. Your prime objective is to push to their tower so they'll lose cs, experience, and vision of what you're doing. However, when enemies try to freeze the minion wave outside their tower range, continuing to farm will only leave you susceptible to enemy ganks. As such, pretending to leave lane will leave the enemy to push the wave because they will have thought you went back to base. When enemies leave the lane it's best to push to their tower so that you can force them to lose CS. Also, when pushing heavily reminding the support to ward further away will help you get better vision of enemies incoming to gank. Always aim to push the enemy tower down, and when you do so, you can choose to roam mid lane to gank or push the mid lane turret down, or farm the jungle (potentially red buff). Pushing the tower down will give you a lot of leniency because you will not be forced to stay at bot lane. However do go back down when the wave is pushed to your tower because you don't want to miss out on the experience and farms.

Teamfights
As an ADC, you will always want to stay in the back line during teamfights. However, prior to fighting, whenever you can without leaving your zone, aim to poke enemies down with Mystic Shot so they have a larger disadvantage. However when enemies are obviously wanting to engage on you, back out. Positioning is relatively easier on Ezreal because once enemies dive you you can just use Arcane Shift and start kiting again. Kiting is extremely easy on Ezreal if you choose to get Iceborn Gauntlet.
You will always want to be near your tank line while attacking anything you can. Often times your main focus will be the tanks and bruisers because to focus carries you will need to leave your safe zone and get in range of enemy tanks and bruiser who could easily CC you down. Use your Trueshot Barrage either before a teamfight you know would be tough so your passive will be proc'd or after a teamfight to snipe down running enemies.