How can I effectively use Genji's skillset?

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.

Genji fulfills the same role as Tracer, his strengths lie in flanking and guerrilla tactics. Don't fight from your team's frontline like you would with Pharah or S:76. Take alternate routes around the map and target the enemy team's backline. Occasionally, harass them as they try to travel from spawn.

Between wall-climbing, double jumps, and his dash, Genji's raw mobility is unrivaled. Tracer may be more agile in 2D, but she is largely bound to the ground. Pick and choose your fights because you should be able to escape any that aren't in your favor. Don't be averse to using your dash to run away. Be sure to target enemies with low health that you can quickly execute because that allows you to reset your dash and escape.

Also, don't overstay your welcome. Genji's main advantage is that his mobility allows to him attack his enemies while they are focused on your team's frontline. As soon as you grab everyone's attention, you need to run because even with his Reflect, Genji is fragile. Don't worry too much about escaping before securing a kill, it's better to live to fight again in a few seconds than dying and having to wait to respawn.

This keys into maximizing Genji's ultimate. While it massively increases his damage, Dragonblade also dramatically reduces his available range, requiring him to engage at shorter distances without increasing his durability. A Genji that attempts to solo an entire team with his ult is a Genji that will be shortly respawning, use his ultimate when the enemy team is distracted or occupied with another threat such as when they are scattered to avoid another ult.

Genji is one of the most difficult heroes to master, it's OK to struggle with him. I'm no Genji master but this video (albeit a bit old) really helped jump-start my Genji game.

Here's a summary of the tips found in the video:

1. Learn the maps and focus on the flank. Use his mobility!
   Use his mobility to attack enemies from the side or from behind.
2. Prioritize targets, go for backline squishes!
   There are a lot of heroes Genji shouldn't be fighting because he'll die rather easily. The tanks are one example but avoiding getting close to Mei, McCree or Symmetra is also essential. Focus on attacking snipers and support heroes. You can also take out Bastion.
3. Maximize your damage output (proper use of your abilities).
   Lear your shurikens. They're tough to land and they don't do that much damage. Practice, practice, practice!
   Use combos: get behind someone, use alt fire a couple of times, melee him into Swift Strike for instance.
   Ultimate: use it when the enemy team is distracted by your allies and you're sneaking up behind them. Do not run up to your enemies from the front with your ultimate activated because it's almost certain they'll all focus on you and you'll get killed rather quickly.

The video contains the basics of Genji only but it's really helpful.