You'll want to spend two nights in the canyon. Maybe three if you can get them. A single overnight makes for a very strenuous hike, as the second day is very long and very steep and very relentless uphill. You'll probably be happier going south>north; make your last night at Cottonwood, which gives you around 7 miles uphill on the last day. Take your time heading uphill, and try to do it very early in the morning.
We hiked from the North Rim in late June a few years ago — it was 37F at the parking area, and 105 at Roaring Spring. So be at least a little prepared for cold weather, though it's unlikely you'll need to camp/sleep at very cold temps.

