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Rainy season (tsuyu) doesn't affect the whole of Japan in the same way. Hokkaido usually escapes it completely (it can of course rain from time to time, but not a persistent rainy season as such).
Usually during tsuyu the sky tends to be grey and you get steady rain on or off. It's not like the tropical monsoon downpour - you normally only gets it with a typhoon, which can attack Japan in July. What makes this season particularly unpleasant is the very high humidity, which if you aren't used to it, can make any sort of outdoor activity very unpleasant. Avoid synthetic clothes and stick to cotton or linen. Shorts are good and locals wear them a lot, except where business attire is expected like in an office or bank. You do get occasional sunshine, but if you expect grey days, you won't be too far out.
Tsuyu officially finishes anytime from end of June (in Okinawa) to middle to end July (in Northern Honshu). In the Tokyo area, it's usually around the middle of July, but last year, when I was there, it ended early, around 10th and it soon got very warm - up to 35C and strong sun, and still with high humidity.Take your own favourite sunblock. It's not so widely available or cheap.