I travel to Spain 2 or3 times a year to visit the wifes family. Whilst were there we always take the time to do some hiking, mostly in the Sierra del Sur (about 50km north of Granada), where her family live, and the Alpujarras. However, we have also hiked in the Picos. Generally, although the scenery is different the way things are setup is very similar.
Hiking in Spain is popular, but the Spanish being the gregarious types love to do it in huge gangs. Whilst there are marked trials in the national parks, these tend to be along well used tracks and paths that will take you to the highest peaks and loveliest views, but unless you travel in low season, you will not be alone.
Travelling across country can be done. Topographical maps at 1:25000 are available, but they are the military type with little in the way of landmark detail. Find out where they sell them in advance, because youll probably need to make a special trip to buy one or get them sent to you in the post before you leave. One thing the maps dont mark are the fences. These range from designs to keep the sheep in, to 3 metre high barbed-wire topped jobs. There are a lot of them in general and they can be very annoying, blocking 4x4 tracks that you think would be perfectly fine to walk on. The reason for them isnt to keep out foreign hikers, but to protect the hunting rights from neighbours - I recommend that if youre going off-piste, you make sure you have a bit of fluorescent orange rag tied your pack.
I have camped out a couple of times, and the only people Ive met have been the shepards. They tend to be incredulous youre there (why would you sleep on the ground when you can go home and sleep in a bed?!), but really friendly. Bear in mind that *nobody* in the countryside will speak a word of English.
Obviously, water can be a problem from May until October. Water, like hunting, is jealously guarded (again from neighbours, not bushcrafters), so plan ahead and find out where the year-round springs are.
Itll take a bit of organising, but it can be done.