If you want to go AP, Mana Crystal/Doran's Ring start. Build Tear of the Goddess first. Next, build Sorc Boots, Mejai's Soulstealer, Sheen>Lichbane, finish Archangel Staff, Guinsoo's Rageblade, Zhonya's Ring.
For AD, rush Black Cleaver asap, Ninja Tabis/Mercury Treads, Frozen Mallet, everything else is up to you. I never play AD. AD is better on TT, but AP is better on SR.
Here's my ezreal, build him like this almost every game.
Start with Sapphire Crystal/2x Health Potions, try to get solo mid if possible. Make sure to get your minion kills, hold position and wait until you can one shot them, the rest of the time you should spend harassing whatever mid champ you're against.
Build your sapphire Crystal into a sheen, grab boots. Build your boots into either mercury's treads (if their team is stun/magic heavy), or sorcerer's boots if you don't need treads.
If you find yourself destroying the competition, build a mejai's after boots. If you're finding your team is getting destroyed and you're dying too fast, try to build a Rod of Ages.
Turn your sheen into a lich bane.
If the game is still going build a Rylais or Zhonyas. If you're getting toasted you might want to grab a banshees veil.
Runes should be = Red = Magic Pen Yellow = AP per Level Blue = Reduced Cooldown per Level Quintessences = Reduced Cooldowns per Level
Bring Flash and Exhaust or Flash and Ignite, I find these are the best for Ezreal. You might also try teleport to get back to your lanes faster and assist around the map.
Forgot to say, being able to anticipate enemies movements and hit them with your abilities is the biggest part of playing a good ez, and staying alive. Map awareness as well, ezreal is a great ganker. If you see a hero run away to try to recall next to a tower, use your E ability to teleport near him and hopefully grab an easy kill. If you're good enough, you can get a lot of kills from across the map with your ult. I always max flux first, followed by E, then Q. Your Q ability doesn't really get that good until you get lich bane, then it destroys. I usually leave it at level 1 until everything else is maxed and I have my lich bane.