Even thought On-hit Ezreal is not the BESSST thing ever, it is probably one of the funnest I have played in a long time. As an ADC Main I loved playing Ezreal and I once accidentally built him wrong. Instead of getting a Tri I got a Infinity Edge and Stattik :P thinking i was playing Caitlyn or something... Anywho On-hit Ezreal is made to do not a lot of damage but have fun.
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Ap Ezreal is AWESOMEEEEEEEE. Ez's 'w' does crazy damage and his ult can half hit people. Its truly amazingly fun.

You might be wondering: Why build Bork?? Well I got an answer right here for you. You might not know it but the lifesteal procs off Ezreal's 'q' which is mainly the point of On-hit. Stattik's electric can crit and is also very powerful on Ez's 'q', which can also crit. The Last Whisper is a very interesting item, thus it's Unique Passive Armor Penetration procs off Ez's 'q' which makes armor not a real threat. Iceborn and Muramana just makes stuff more fun. I also got Blue boots because I like to have my 'q' on a low cooldown.
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Get a Mejais cuz stacks. The rest of the items are pretty self explanatory, Just do damage and win.

I usually get a Tear and a Sheen first back so that I can spam 'q' and stack the Tear. Second back if im fed I get a BORK or if I don't have enough gold I get a cutlass. Around this time you should usually finish blue boots and be moving on to the Stattik. After Stattik I would complete my Manamune and Iceborn, then finish off the build with a Last Whisper.
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When Winning lane get mejais, when losing lane get mejais then roam. Thats how you play AP Ez.

When CS'ing, I would usually back at 10:00 to get a sheen and a tear, maybe even boots, but usually I would just CS with my AA's and my 'q' if the creep is out of range. I usually would go all in with AP Ez but with On-Hit, you don't want to go all in. You would like to just use your 'q' and 'w' to poke, but when you clear their wave quicker you can 'e' 'w' 'q' to deal maximum damage.
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Same as On-Hit, get tear then spam 'q' to farm. Poke with 'w'.

On-Hit Ezreal
Pros:

-Fun
-'q' does a lot of dmg
-Can spam 'q' after getting manamune

Cons:

-Not as powerful as AP Ezreal
-Mid-Late game build

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Ap Ezreal

Pros:
-Tons of Damage
-Very fun to play


Cons:
-Squishy, so build armor and mr
-Not as strong early game than mid and late game.

Usually as ADC I would get Flash and Heal, but sometimes you can switch out flash for something else, like clairvoyance or ignite. Only if you want to because your 'e' is basically an 18 second flash that does damage. I would always keep either flash or heal, which is usually what I go as ADC
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Get Clairvoyance because of it low cooldown and you can crossmap snipe people with your ult.
Nuff sed.

When trying to hit your skillshots as Ezreal, You usually want to just anticipate their movement, like playing Thresh and Blitz. Maybe even do some madlife stuff.... :P Your Ult does not do as much damage as AP Ez but you can use it for execution or pulling off some sick cross map kills. I usually 'e' to my left or right, never straight infront, then 'w' 'q' after I 'e'd.
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Your 'w' does a lot of Damage, so does your 'q' and 'e' but not as much. Poke with 'w' and not 'q'.

In teamfights, You want to keep your Ult and just poke with your 'q' and 'w' from a range, then if your chasing you can use your 'e' or when your running you can use your 'e'. Remember that your 'q' now does AoE damage from the iceborn, so you can just spam it in the teamfight because it also procs Stattik.
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In teamfights, as AP Ezreal you usually wanna just stay back and to the sam as what you do with On-Hit Ez, You just wanna poke with your 'w' and 'q' and finish people with your ult.