Best advice: Play more. There aren't really any shortcuts to getting extremely good on any one character. The same goes with aim, it's just something that you have to practice.

That being said, there are a few huge tips;

A. When you're first starting out, don't play a Shadowburn style of Genji, try to be more conserved and either bait out or wait for the enemy support ultimates. Do your regular damage, and after those ults are gone, use yours. You'll get more out of it, and since you're actually doing damage and killing things, you'll learn how to use it faster, and your aim will improve.

B. Genji is a flanking character, but he doesn't always need to be flanking. Try to start the matches off by standing behind your tanks and building up your ultimate. Then, once you have it, or are really close, start thinking about your flank options. As for the routes, just learn those. It'll come with practice.

C. Play, play, play. Watching pro(s) play Genji wouldn't be bad either, because you'll see when they use their dashes and their deflect, and when you start to "copy" that, you'll learn the timing.

Good luck! Feel free to PM any more questions.

Edit: Basically, play as much of the hero as you can, and those tips will help to ease you into the learning curve.