Playstyle depends on our support. Either harass the opposing AD Carry constantly in order to stop him from last hitting or we passively farm. In team fights a good AD carry needs to have good positioning, which can win a team fight. The best place to be is where enemies can't hit you but you can attack freely.
Don't forget W passes through minions, meaning the opponent can't be safe even when behind his minions.
When we can, we harass with Q, gaining extra attack speed which gives us better trades. Carefull in spamming W, it can run you out of mana VERY fast, something that the opponents can take advantage of.
Tips & Tricks
Ezreal's role is like all AD carries , do as much damage as possible to enemies while staying alive in team fights. Never ever use the E in order to engage a fight or use it to deal damage, must always use it in order to stay safe from anyone coming for you.It can be used to go through a wall and force the opponent to use his flash in order to catch you. Finally with Trinity Force u can easily kite an enemy due to the high mobility it offers.
Graves: Strong opponent due to his passive, giving him armor and magic resist. Don't fight him in close quarters and be wary of his burst.
Vayne: Not a strong laning phase AD, but she becomes a monster late game. Get really agressive on her, prevent her from farming and try to get an early lead.
Corki: Hardest lane for Ezreal, because of his burst damage. You need tankiness from a couple Doran's Blade. If we have a sustain support try to do a lot of small trades. If not, play passively and farm.
Ashe: Easy lane as W gives us an advantage on trades. Dodge her W by staying behind minions.
Kog'Maw: Same as Ashe, just avoid trades when he activates his W.