Was about to type something completely different, but then realised you were talking about a road trip.
I can't offer any advice about Finland, but Luleå to Nordkapp by road could make a decent trip, and plenty of people do the coastal Norwegian part of this as part of various routes (see examples above - Crazybrab speaks sense). Narvik-Tromsø-Alta must be amazing if you were to ride it in good conditions. Going through the interior, Kautokeino to Alta would be totally different and amazing in its own way if you were lucky with weather. Distances between things are big (probably obvious) and the landscape can be pretty relentless (or bleak if you're not into it). Sticking to the coast takes you through lots of towns/villages - it's not as sparsely populated as you might imagine. Away from the coast is a different story however. Culturally speaking its a pretty interesting part of the world - which many might overlook (not surprisingly). Wildlife-wise, you'd be lucky to see anything scarier than elk and reindeer but lynx, wolf, brown bear, wolverine and so on are around. Take a small fishing rod and you won't starve (along with berry/fungi foraging if you're there in late summer). Bike wise, I'd take something you are comfortable riding for sustained distances and reasonably efficient, reliable and with robust wheels over an out and out road bike.
Getting to the start points, you could easily use sleeper trains from Göteborg or Stockholm to get to Kiruna or Gällivare (www.sj.se for details), or internal flights on SAS or Norwegian to any of the many local airports in the region via Stockholm, Oslo etc. Either way I certainly wouldn't reckon on paying more than a couple of hundred quid return from the UK to get there and back. Personally I prefer the train, but if I went via Norway I'd fly.
If it were me doing it, I'd probably start in Kiruna or Gällivare and do a loop doing both interior and coast with a small detour into the Lofotens. The prospect of riding to and from Luleå doesn't add much to the trip for me personally, unless its the idea of joining two coasts - in which case the interior route is 'purer' to me. Now (at the risk of hijacking the thread) if anyone has any further off road suggestions in the region, I'd be interested to hear.
*should add that I'm assuming you were going to camp (free, flexible, safe).