Hanzo is one of the trickiest heroes in Overwatch and is far from a good choice for a new player. Part of this has to do with his tendency to play as a sniper, like Widowmaker, but with a bow that requires compensating for distance due to the drop in velocity of his arrows. Timing head shots while also managing the natural arc of an arrow requires practice, but fortunately Hanzo has some good utility to help make him feel more useful. If you like sniping but want an alternative to Widowmaker, Hanzo is worth a try.
Storm Bow [LMB] - As mentioned, Hanzo's storm bow is an inconsistent and unwieldy weapon that often feels outclassed by the likes of Widowmaker or even McCree. Holding LMB will charge the shot, making the arrow fly faster and reduce its drop but also slows Hanzo's movement speed and rate of fire. Firing without charging the bow makes arrows fly slower, do less damage, and you'll need to compensate for their arc much more.
Scatter Arrow [E] - Scatter Arrow will take a lot of the pain out of landing shots with Hanzo's Storm Bow. When this ability is used, Hanzo fires an arrow that, upon impact with a surface, explodes into smaller arrows that bounce around. This makes Scatter Arrow extremely useful for clearing out enemies in tight spaces or those hiding behind obstacles.
Sonic Arrow [LSHIFT] - Sonic Arrow adds a ton of utility for Hanzo and should be used as often as possible. When used, Hanzo fires an arrow that sends out a sonic wave that highlights any enemy caught in its radius for your team to see. It's a great way of giving your team intel on enemy movement and, used frequently, can be more useful than Widowmaker's ultimate which does something similar.
Wall Climb [Space] - Hanzo's Wall Climb let's him scale up surfaces to reach high ledges and platforms to get great angles for sniping opponents. There's no cooldown either so Hanzo can get from place to place much easier than, say, Widowmaker.
Dragonstrike [Q] - The crown jewel of Hanzo's abilities is Dragonstrike. Hanzo fires an arrow that, after a short distance, spawns two spectral dragons that encircle one another in a massive column, damaging anyone they happen to touch and passing through walls. Dragonstrike is great for punishing entrenched enemies like Bastion or Torbjorn, but also acts as a great way to push. Everyone on the enemy team will scatter, creating an opening for your heroes to rush in.
Hanzo Strategy Tips
Hanzo's high skill ceiling will make him unpopular with most players, but he's also one of the more satisfying heroes to master. In essence, he's a more aggressive sniper than Widowmaker, meaning you should always feel comfortable getting closer to your targets—which also helps negate the travel time of your arrows. If you can, always charge a shot fully before firing, the damage output is considerably more and will kill plenty of heroes in one hit with a well placed headshot.
Scatter Arrow should be used to flush out enemies like Junkrat who exploit cover while still harassing your team. If another hero gets a jump on you, it can also be used in a pinch to save your bacon as long as you're able to place the shot appropriately to ensure your would-be murderer takes the brunt of the arrows (hint: aim at the floor!).
Your Sonic Arrow should also be firing as often as you can, but knowing where to place it is going to take some practice and knowledge of the layout of each map. If playing on defense, it can be used to show your team which door the attacking team will be coming out of and also what heroes they'll be using in the opening minutes of a match. Otherwise, use it to give information to your team before pushing onto the objective when attacking or to discover an attempt to flank your position. In the Control game mode, however, Sonic Arrow isn't likely to get much use since everyone will just be congregating on the capture point.
Dragonstrike is often seen as a devastating finisher, but its usefulness is more nuanced than that—knowing this will make you a much better Hanzo player. While killing heroes is great—and you surely will—think of Dragonstrike as a train that everyone is going to want to get out of the way of. When on defense during Escort modes, this can clear enemies off of the payload and give your team precious seconds they need to regroup. Its ability to pass through walls will also be useful when a Bastion or Windowmaker has setup on various maps to snipe your heroes from behind like during the second phase of the Hanamura map.