Ezreal was the first champion that I played ADC when i started to play League again. Even though at first I was doing really bad with him, I thought it was so fun to play him. That is why I still play Ezreal. I will still be adding to this guide, because I am still learning what best fits Ezreal and what potential he has.

Pros
Very strong mid game with two item power-spike.
Good late game.
High mobility.
Times spent playing this champion significantly rewards you.
Fun to play!
Cons
Generally weak early game.
Skill-shot based.
Takes a while to getting used to.

This runes are pretty basic for Ezreal. One possibility is to change out some magic resist runes with scaling cooldown reduction or scaling magic resist runes, which can make your late game stronger.

Many of the masteries are basic, but some changeable ones are the Fresh Blood and battle trance level. On the Fresh Blood level, Fresh Blood and Feast may seem about equally good, depending on you play style. Feast has good sustain, but Fresh Blood has good poke. Take Fresh Blood. The damage highly outweighs the sustain. If you proc it two times, it has more health advantage than feast. They cool-down is also five times shorter, so you will be able to do lots of poke damage every 6 seconds.
Vampirism gives you life steal, and natural talent gives you scaling ad and ap. All of Ezreal's abilities use ad and ap, except essence flux, which only uses ap. The ad and ap both give you extra damage to your attacks and abilities, so it is much better than 2% life steal. Plus, you already get 3% life steal from Doran's blade. I take 1 point in life steal just to get a tiny boost in healing.
Battle Trance give you 3% bonus damage, increasing over 3 seconds. Although that is slow, there is no penalty. For Bounty hunter, you have to get a kill first to make it useful, and you need three different champion kills to get to three percent. Double edged sword gives you flat 3%, but you take 1.5% more damage. That can be pricey when the enemy team is focusing you. Take either double edged sword or battle trance, usually battle trance. Battle trance has damage increase and doesn't increase your squishyness, so overall it is good. Double edged sword has better poke, because you aren't in combat for a while.

Doran's Blade
Best Starting item for ADC's by far. Gives health, damage, and a little life steal. Also get a biscuit for staying in lane longer.

Tear
Ezreal's key item. He uses a lot of spells, so this is a very good buy. When going back, extra gold can be used on boots, refillable potion, or cull. I usually try to get boots for the extra mobility against your opponent.

Sheen
Sheen is a very good item for Ezreal because it procs of his Mystic Shot, or q. His q is where most of his damage comes from.

Vamperic Scepter
You may want to get this if you are having a tough time in lane. If they keep bullying you, you can get this item for the extra sustain. Then you would rush Blade of the Ruined King, or BOTRK, instead of getting Manamune.

Manamune and Lucidity Boots
Manamune gives Ezreal mana and damage because the upgraded version proc of his q, dealing a lot of damage. This is a must for Ezreal. The Lucidity boots are also very good because Ezreal just wants to use his abilities, so the cooldown reduction is very good.

Sheen Upgrade
Trinity Force, Triforce, and Iceborn Gauntlet, Gauntlet, are the two sheen items that Ezreal uses. Overall, Triforce is better, so it should be your go to. Gauntlet is situational. If they have a lot of attack damage or assassins, then go for Gauntlet because it gives you cc and armor. Also if you haven't already gotten BOTRK, then get it. It is very good and the passive procs on your q, dealing massive damage.

All the other items
It all depends on what the enemy team has. If they have ap damage, get Maw of Malmortius. If they have a lot of cc, get Mercurial Scimitar. If you get one-shot by the enemy mid laner, get guardian angel. They have armor, get armor pen. If you just want more damage, get a blood thirster. Adapt to the enemy team.

Pretty basic skill sequence. You want to max R>Q>E>W. At level four though, you want to take W instead of E. W can gives you more utility, and getting E just does a little more damage and less cool down.

Very basic. Always take flash and heal. Flash allows you to escape death, and heal allows you to live longer in fights, so you deal more damage.

Don't underestimate Ezreal's passive, Essence Flux, and Arcane Shift.
In team fighting and regular fighting in general, you hit many spells on your enemies, procing you passive. Your passive can give up to 50% attack speed at level one, which is more than any attack speed item. That means you will be attacking really fast, so don't forget to auto attack a lot in fights. Essence Flux can hit multiple champion target. This does a decent amount of damage at low levels, but most importantly procs your passive very fast. If you hit it against your opposing laners in a fight and hit two mystic shots, you already have 40% attack speed. In a team fight, using Essence Flux first can get you passive to almost max, just hit a couple Mystic shots, and then you have 50% bonus attack speed. Arcane Shift does damage to the nearest enemy target. When going all in, e next to the person you are attacking. Your e damage will hit them and your skill-shots will be easier to land.

Ezreal is a good champion to main. If you can get very good at his skill shots, it can be very rewarding. Landing all your skill shots when fighting down bottom means that you will almost always win the fight (unless you are in a really bad position). I wouldn't recommend playing him if you don't play him that much. He has a pretty low win percentage compared to other adc's, but is one of the best carries if you know how to play him right.