Hammerfest is the ideal city to experience the northern lights in the winter and the midnight sun in the summer. From the 16th of May to the 29th of July you have the chance to experience the sun that never sets, the midnight sun. In summer it never gets dark! But in the autumn it gradually gets darker and darker, and from September the first northern lights can be visable. From the 22nd of November to the 20th of January the sun never rises above the horizon in Hammerfest. We call it the polar night. During that time it's almost dark all day, but around noon you can experience "the blue hour," before the northern lights takes over the show.

The northern lights are almost always in the sky during the winter. The main thing stopping one from seeing it is the clouds. Since it is so dark during the winter it is hard to tell if it is cloudy or not. There are several websites and apps with northern lights forecast. 

Tips for photographing the Northern lights
Capturing the northern lights on a camera can be challenging. A phone with the standard settings will not be able to capture the bright light in the sky. Instead you will get a black picture with some vague lights showing. There are a lot of guides online for capturing the northern lights on camera. If you only have a phone to take pictures with you can search online for the best settings for your specific phone.

This is some recommended settings you can use on a camera:
Use an aperture of f/2.8 or the widest in your lens
Adjust an ISO from 3200 to 6400
Set a shutter speed between 1-15 seconds
Adjust your white balance to 3500k
Focus manually on a distant light
Set the general camera settings for Northern Lights
Select a shutter delay of 2 seconds
Check your Northern Lights shot histogram