If you want to see the Northern Lights, you have to go to the north (from Tromso upwards let's say) in the middle of winter. It's cold! But there are also organized tours in the woods.
If you want to organize a "do it yourself" trip by car, van or camper, the best time is from June to mid-August, when the weather is milder, it doesn't rain a lot and the days are very long.
When wandering around northern Norway, pay close attention to reindeer! They often walk on the roadsides regardless of cars.
In Norway you can camp freely anywhere, just don't be too close to the houses and leave everything clean.
When you arrive in North Kapp, if you go after 10 in the evening, the parking (otherwise very expensive and unavoidable) is free because the whole structure closes, but you can safely pass the controls.
In Alta there is a beautiful museum of rock art with graffiti dating back up to 7000 years ago! There is an outdoor path with a beautiful view of the fjord.
The most beautiful camping area in the north where I went was Ersfjord, not too far from Tromso. There is also a bathroom (one only for the whole beach!) Where you can take a shower.
An area not to be missed, very close to Tromso, is Sommarøy. A fishing village worth stopping by. Ferries leave for the Senja peninsula not far away.
In Lofoten, not far from Svolvaer, there is a place known among climbers, it is Kalle. You will find climbing groups and you can ask to join, maybe someone even has equipment to lend you! There are public toilets and cheap showers.