I walked around the same time a few years back. Walking from SJPdP on the first day, I experienced rain, hail, and snow; the next week was almost non-stop rain. The second half of the trip, though, I had blue skies nearly every day.
Looking back, I was probably unlucky, as many people point to May as an ideal weather month. However, be prepared for a greater range of possibilities, especially in the mountains.