In reply to subtle:
Castles - Alnwick castle would get my vote over bamburgh, which is only good for the view, but you can get that for free from outside.
Farne Island trips are only £15 - £20 and well worth it, but I'd watch out for the weather this time of year. If you think its exposed on the beaches, then it can be particularly challenging on the water. The boatmen are all ex (or even current) fishermen and dont really mind about a bit of a swell. You may think differently.......
Beaches; subject to the wind direction, all are bloody brilliant from Blyth up to Berwick. Embleton bay with dunstanburgh castle in the background is pretty special. And empty. Then go to Ross sands, which will be emptier.
Pubs / restaurants everywhere, especially pubs. Greys Inn at Embleton gets my vote for beer and hearty meals if on the coast (its a bit of a weekend local, so I'm a bit biased) , and the cheviot inn in Bellingham if near Kielder for exactly the same reasons.
Climbing - what the others have said.
Biking - theres a cycle path gores all the way down the coast, which is really nice, and mainly off road. Or go inland to Kielder reservoir, and fill your boots in the forest.
Why go for a week? Try all the above, then book another week to go do it all again.
If you arent familiar with the lansdcape, have a look at the "Hidden Northumberland " facebook page.