The most beautiful manner of getting to Hammerfest is probably to come by boat.
You can take the coastal steamer Hurtigruten. Most of the Hurtigruten ships can take cars (but no caravans) and come to town twice a day – one southbound (10:45) and one northbound (05:15). Northbound you can get to the North Cape within 6 hours from Hammerfest. Southbound, the route leads to Tromsø and arrives there around midnight.
There is also speedboat connection between Alta and Hammerfest. The speedboat goes twice a day from Monday to Friday and on Sunday once. There is no boat on Satuday. Find departure times here or download timetable.
Every year various cruise ships from all over the world visit Hammerfest too. If you would like to come with your own boat, please have find some basic information here.