You should be aware of the distances involved. By far the easiest way to get there is to fly. Use the travel planner of your choice, and check the websites of Norwegian (a somewhat low-cost airline with less draconian luggage practices than Ryanair) and SAS (the flag carrier) to find out when it is cheapest to travel (apparently there are no direct flights from Berlin to Tromsø). If you have a rail pass, you could also consider taking the train to Stockholm or Oslo and fly from there. The Tromsø airport website has a list of direct flight connections.
By train there are essentially two options: to Oslo (by ferry from Copenhagen, Kiel, or Fredrikshavn, or by train, changing in (at least) Gothenburg) and then onwards to Trondheim (ca 6h), change trains, to Fauske near Bodø (ca 10h), change to bus, and then a ca 12 hour bus trip (seems long, Google maps says 7h driving time, might be less than 12h. But includes a ferry connection). Or to Stockholm, onward to Narvik (ca 13h) (nearly all the train travel is inside Sweden, Narvik is not directly connected to the rest of the Norwegian rail network), and then around 4h by bus to Tromsø. The connections might not match up nicely.
Some brief information from the Tromsø tourist information: http://www.visittromso.no/en/Articles/Tromso_useful-info/Travel-by-bus-and-train/