Hi, Jenny,
Oh you'll love it. If I could choose any time of year, I'd make it early fall (Sept-Oct) or early summer. I think June is the perfect time of year for the Primitivo. Temps are warmer, but there aren't huge crowds. My impression is that the only time the Primitivo really gets crazy is August. I was there in mid-late June last year (arriving Santiago around June 26 or so), and we had enough people to cement that pilgrim feeling, but no crowds. And doing the Salvador right before then was also great weather, snow gone, flowers in bloom (but no other pilgrims).
I usually walk in May, but I don't think there are any disadvantages to pushing back the Primitivo till June. And that gives you a little more likelihood of better weather. We had one all-day rain, but it wasn't that chilling cold rain that makes you shiver.
Buen camino, Laurie