More suggestions than guide, as I don't play Hanzo all that much and got this back when it was for Scatter Arrow.
Edit: It's been said that you don't need to get the final blow, but do damage to the enemy. So, do try to do most of the damage, or some damage, pump into team fights and pray.
Storm Arrows is a rapid fire ability. The arrows do slightly less damage, but are fully charged, so they will travel the full distance. It charges 6 (six) arrows that you can fire whenever within a 6 (six) second window. Holding down the trigger shoots them in rapid fire, or tapping the trigger shoots them one at a time.
Try to use this ability in choke points or team fights, where enemies are weaker. Look for squishier targets, people with under 250 max hp, as 2 body shots should kill if they are damaged. These kills do not need to be solo, nor do they need to be headshots. You just need to see your name in the kill feed as the killer. 2 headshots does kill most squishy heros, so if you're a god Hanzo, that is an option.
Edit to add: immobilization ults like Rien, Zarya, or Mei could be of use as the enemies will be stationary for a few seconds. Pump some arrows in and try to get some kills.
Again, look for choke points and team fights. Let your teammates soften them up, then try and land the final blow. Squishies are easier to steal.
Good team fights locations I can remember. Point B on Anubis, the courtyard before A and on point B on Hanamura, and King Of The Hill map's point, lunar colony A and B, Volskya A, Hybrid map's point too. These are suggestions, and team fights can happen in many different places on many different maps, these are just maps I see choke points and team fights more often. Players in quickplay seem to bunch up in these maps more often. Just keep an eye out, no matter your map.
If all else fails, try and join a boosting session. It's not guaranteed to get a lobby, but it is significantly easier to coordinate for 3 kills.
Good luck, and good hunting!