I do pretty **** well with Ezreal, and I got to 2150 before Jabebot died so hopefully my advice will help you too.
As far as positioning is concerned, you want to stay back. Way back, always make sure one of your teammates is ahead of you leading the charge. Ezreal can be pretty fragile when caught, cc can make it really hard to kite (of which Ezreal excels at tremendously) so keep your head down and wait for your moment. This pretty much goes for all ranged carries, but with Ezreal this is much easier as you have Mystic Shot to poke with while you wait for your initiator to jump in to give you an opening.
How to use Arcane Shift
This is the most common issue I see new Ezreal players doing. You hardly ever want to blink in offensively, the only time you should ever do this is if you need to close that tiny bit of gap to certify a kill, and even then only do it if you know you will get out alive. I see too many people trying to use it to add more to his burst, but it's not a good idea because it's your only escape, and the damage on it isn't even reliable. Always, ALWAYS use this to further your kiting ability. Ezreal is amazing at kiting opponents, so make sure you always have this on hand to jump out of a fight if you need to, not to jump into one.
How to use Essence Flux
You typically want to use this on an enemy carry as the attack speed slow will benefit your team more than the attack speed bonus you give to your team. However, this goes into positioning. You can technically hit both at the same time, but I wouldn't go too far out of my way to hit both. It's gotten pretty subconscious after a while and you get used to placing yourself in positions where you can hit more than just one target.
AD vs AP
Both are viable, but I wouldn't say one is superior to the other. Both playstyles are very strong in their own ways, and have their own downsides. But they are just too different to really say which one you should always go for over the other. Hybrid is even good too, if a bit more expensive than the other builds. Just please god don't fall into the common noob traps that so many Ezreals fall into (lucidity boots, starting sheen, etc.). Can they work? Well, Ezreal is a freaking monster on Dominion so I'm sure whatever you do CAN work to an extent. But the other options out there are just so much better in so many ways.
I go 21/0/9 masteries with arm.pen reds, health/lvl yellows, mres/lvl blues, and movement speed quints. Sounds weird, but it works really well for my ranged carries on Dominion, especially Ezreal as you are doing near true damage unless they spec into armor early when you open with Brutalizer, and it gives you a healthy amount of defense to keep yourself alive throughout the game. For spells, I typically take Ghost/Exhaust, but Revive is a good choice too. Flash is a good choice on him too, and I know that spell isn't very popular on Dominion but it will save your life a LOT if you use it efficiently.
For the record, Ezreal can bot but he'll lose to a lot of the more common match-ups. Champs like Malzahar and Yorick especially can be very difficult to deal with because their minions will constantly block your skillshots from taking down creeps. But really he's just way better top lane either way, there isn't a good enough reason to take him down from top where he does so well to put him bot where he is basically just mediocre.
I also recommend smartcasting. Takes a little bit to get used to, but my damage skyrocketed when I did it as I can weave my spells between auto-attacks way more efficiently and I am way more accurate overall with my spells. It's not for everyone, but I do recommend it.
If you want to add me in game, invite me and I'll play with you. I won't promise that I'll do amazingly as Ezreal as it's been a little bit since I've used him seriously but I'd be happy to play through a game with you.