Hi, Marie,
I think the best place to start, to see what the accommodation situation is on the Norte, is mundicamino.com. It's pretty intuitive, click on Camino del Norte, then you'll have a list of stages come up, click on the stage you're interested in, and for each stage there is a list of all the hotels/pensiones (hospedajes) as well as all the albergues. Many of the places listed there include links to their websites so it's pretty easy to find out information about location, prices, etc.
I have stayed in different private places along the norte for the stretch that you describe.
In Bilbao, I stayed once stayed at the Pension Iturriena Ostatua, somewhat charming, not fancy, but very centrally located in the historic core.
http://www.iturrieneaostatua.com/
On another occasion, I stayed in Portugalete -- at the Gran Hotel Puente Colgante, a decent place, nice views, comfortable.
http://www.granhotelpuentecolgante.com/ -- it wasn't too pricey then maybe because it was off season.
In Castro Urdiales there are plenty of choices because it's such a tourist town, I stayed in a pretty basic but very well located pension:
http://www.lamarcastro.es/
In Islares, we stayed in the Hotel Arenillas, where we had a fabulous meal in their restaurant. We spent the afternoon walking down to the water, it was very nice. Hotel is not fancy, but the restaurant is hugely popular, at least it was on the sunday we were there.
I stayed in other hotels on this stretch but I can't remember their names. The thing about the Norte is that albergues are not as plentiful, so unless you're a careful planner, you'll wind up in hotels like I did. I'm not complaining, because we stayed in some very nice coastal towns with lots to enjoy.
buen camino, Laurie