There is an entire sub-forum here for the Camino del Norte.
I did the Norte from Castro Urdiales to Santiago in September. Weather was great - only two wet days.
I did Irun to Vilalba last year in July, which apparently happened to be a very wet July, I was so happy to have my poncho. This year I did Avilés to Santiago in June, and it was cool and occasionally wet. The summer months are busier and it can be harder to find accommodations because the northern coast is also a popular tourist destination. I'm planning to do the Norte/Primitivo combo next September/October.