I researched this a few years back, and it's still on my bucket list. I came to the conclusion that I cold either head north through Sweden, or even the Baltic states and Finland, and return via Norway. The scenery is nearly all in Norway, but to be honest the whole thing would be a trip of a lifetime. One thing that did emerge was that I think it would be realistic to allow at least 10 days to cover the length of Norway to the North Cape, given winding Norwegian roads and strict Norwegian speed limits. You could probably make better progress through Sweden/Finland, though. I'd expect an absolute minimum of three weeks for the round trip, and maybe more like a month with breaks for R&R and simply enjoying the scenery. I've driven up part of the Norwegian coast, from Bergen to Alesund, and it's like the west of Scotland written on a much larger scale. I've also driven from Stockholm to Jonkoping in Sweden - you gotta like pine forests!
Obviously if you just wanted to cross the Arctic Circle, you don't have to go all the way to the North Cape, which would shorten the trip by a few days.