Adirondack Park loop is nice, many State Parks with campgrounds along the way with facilities, decent roads, nice views, manageable hills. As previous poster pointed out: ACA has a route. It's around 400 miles, can be more or less if you want to modify it.
I did my variation of it in the beginning last year starting on May 30th and had to cut it short on June 8 due to very cold weather (30s at night). It can be still that cold in June. But it was mu first long tour and I wasn't really prepared for cold weather. I'm planning to do it again a month later this year in the beginning of July.
I have a GPS file with locations of all State Parks with campgrounds in NY State, if interested.