I rode up to North Cape in June 1999. From Bergen up the coast of Norway. Then from North Cape south into Finnish Lapland and down to Helsinki by a straightforward route close to the Russian border. That was about 5 weeks in all I think.
I stayed in Youth Hostals (all in the YHA handbook if I remember) which were absolutely wonderful, and empty, being out of season. And nearly all are in the grounds of an adjacent hotel and you get to use all the hotel facilities for free.
I didn't camp.
I just read the ordinary guide books for the routes and places to see and it was all very good.
In particular, North Cape itself is a disappointment, and expensive. I stayed in the nearby YHA Hostel which was just outside the entrance I think, not too expensive.
But the big recommendation is to walk to Knivskjellodden, a wild headland to the west of N.Cape and actually a few yards further north. It really is a 'must do' visit. There's a carpark at the start of the path, a few miles before you reach North Cape. I set off from there at about 8:00pm on 20th June with good weather that looked like it would hold. I arrived at Knivskjellodden about 00:30am 21st June, and it was truely magical! And dramatic! Completely alone, with the huge orange sun hanging up in the sky to the north and the Arctic Ocean alive with sea birds.
The guide books will advise you about doing this walk, you have to be reasonably equipped. The path is alternately rocky and marshy so its route is not obvious, and is only marked by red paint on boulders every few hundred metres. A hand-held GPS would be good for safety. I just walked in motorbike gear, some water and food, and trusted the good weather would continue.
Another point is that June is mosquito season and you will be innundated with large swarms of them away from the coast. Most people wear proper mosquito hats and head nets. I found lots of deet was OK but a net is better. They say it's dangerous not to protect yourself from them.
All accommodation has special doors, windows and ventilation to stop them getting inside. Close to the coast it's not a problem.
Lastly there's the long tunnel that connects the mainland with the island. It holds some world record for the length of a road tunnel I think. If the weather is fine and sunny when you approach it (as it was for me), stop before the entrance and put on all your winter cold-weather gear. Once you enter, it becomes colder and colder and colder as it dives under the sea. By the time I was almost halfway through I was seriously cold. So I ignored the speed limit to get out quickly, but that's not straightforward as it's very dark inside, and wet, and the walls are just rough blasted rock and not easy to follow once you're going fast.
Have a good trip.