On June 4th, I hiked in Kamikochi with two Japanese, one Brit and one American.
I think May and June is the best season in Kamikochi if the weather is fine (despite the rainy season) because we can see the Japan Alps mountains covered with snow, some wild flowers and the fresh green of trees. Also, it is less crowded than summer.
Especially, this is a season of Nirin-so (two-flower), the botanical name of which is anemone flaccida, one of the rare plants we can see only in some mountain areas. We could see plenty of Nirin-so flowers covering the ground, like a white and green carpet under the trees near Tokusawa campsite.
Tokusawa is a two-hour hike from Kamikochi bus terminal and Kappa Bridge, a famous sightseeing and photo spot. Most hikers in Kamikochi hike between Taisho Pond, Kappa Bridge and Myojin Pond. I recommend that you hike to Tokusawa and stay in the campsite if you like.