The VdlP is very quiet. We hiked it from mid-September through October last year - During the day sometimes we might see 5 or 6 other pilgrims, sometimes none. Including cyclists, the most crowded we saw at albergues was maybe 15. But we didn't stay in alburques in the big cities, so there might be more then.
If you want something even more solitary, we started by walking the 10-day Via Serrana from La Linea de la Concepcion (right by Gibraltar) to Seville. We saw a handful of hikers but nobody who was a pilgrim walking the route we were walking during that entire 10 days. An extraordinary camino, but you have to be resourceful with GPS, etc. - it's not the world's best-markered camino.
I agree with the comments about the time of year. The VdlP gets brutally hot for several months during the summer. But it's a splendid camino with magnificent cities, amazing history and varied terrain.