Hello Ethan,
The best times to see the northern lights in Iceland are probably September through April. The problem is that Iceland is so far north, that during the northern hemisphere summer time, the days are very, very long and the nights are very short. Iceland is mostly below the Arctic Circle, but the days in summer are still very long.
You can play with this calculator to see sunrise and sunset times in Reykjavik, Iceland.
http://www.timeanddate.co...astronomy.html?n=211
Today (March 19) the sun rose at 7:32 am and set at 7:41 pm for about 12 hours of daylight. On June 1, the sun will rise at 3:23 am and set at 11:31 pm for 20 hours of daylight! On June 14, the sun will rise at 2:58 am and set at midnight.
You will not have a very good chance of seeing the northern lights in Iceland this summer, but you will get to experience seeing the Sun in the sky close to midnight!
Kris