My answer is mostly an opinion supported by a few hours of play with Genji. Someone with more experience may be more credible. Key points in bold text.
Genji is fragile, but extremely mobile. His double jump, wall climb, and dash attack allow you to get to positions many other players won't always be watching due to the limited number of ways to access a given point. Open windows and high ledges are your friend. These are typically sniper positions, which you can sometimes utilize as ambush positions or as an escape to disengage a serious foe.
He needs to win each battle with another hero within the first few seconds of engagement. If you're in close, use the right click (secondary fire) of your shuriken attack to release all 3 of your shurikens in a burst, rather than in succession. There are times where the speed difference could matter. If you are harassing at range, plink with left click (primary fire). Be very careful about harassing with Genji, as many characters return fire can cripple or kill you. Always be moving if you can, and be aware of your possible cover positions before you fire.
Engage targets that are isolated. Striking a group is going to get you killed relatively quickly. The exception to this case is having your ultimate ready, which you can readily dispatch groups with the sword, if you catch the party unaware such as from a flank or rear attack. Genji is largely about situational awareness and how to punish holes in defense lines with swift, brutal attacks. If you are striking a stronger character, such as a tank class, you may wish to wait until they are distracted by another attacker as you may not be able to kill them before they react to you. Again, pick targets you are likely to kill, not necessarily ones you might only wound.
If you are spotted during an engagement by a ranged attacker, using deflect will buy you some precious seconds and can even save you from a handful of ultimate attacks. Be aware your deflect is telegraphed visually to your opponents with swirling red glows in front of you and may increase your visibility, despite the client-side visual only showing Genji to be holding his short blade in a defensive posture.