Statistically, the equinox months of September and March are best for aurora viewing. September is a great month in the “Last Frontier.” You can drive, camp, catch some fall colors, and have equal night and day. March would be the best for those who like winter activities.
October & November are quite good, December-thru-February are OK but definitely colder and darker, especially if you plan to travel the Interior (north of the Alaska Range). Please note that, contrary to popular belief, it does not have to be cold to see the aurora, just dark and clear.
April is spring break-up which is pretty slushy and more often cloudy... but that doesn't mean the auroras can't be spotted! The hours of darkness are shorter though.
The summer months of May thru mid-August are just TOO bright in Alaska to see the aurora.
Fairbanks is probably your best starting point for spotting an aurora. Chena Hot Springs, located 60 miles out of town, is a very popular place. They cater to the aurora watcher and they have a fun hot springs and nice restaurant/bar to help entertain the spirit if it's cloudy.
The more independent route would be to rent a car and get a hotel room somewhere in Fairbanks (or beyond). Get a north-viewing room so you can look out the hotel window and check if the auroras are coming in. Start looking for a green band as soon as it's dark. The peak hours are 10pm-2am but I've seen them anytime between dusk and dawn.
If it's clear and you have a “green light” head up to Cleary Summit (Skiland), Ester Dome or Murphy Dome. A 4-wheel drive vehicle would be best for peace of mind. Bring very warm clothing, food, water and a cell phone. If you had car trouble or got stuck it's best to be prepared for the worst because if it's clear it's usually very cold. A parka, face-mask, big boots and those little chemically activated hand warmers would make the difference between a glorious night under the stars versus an “I just want to stay in my warm car/hotel” attitude. I know, I've definitely been there.
Hope is always the key word when I'm on an aurora hunt. Bring books and things to keep yourself occupied because, as you probably already know, there are no guarantees. Pray for luck and luck comes to the patient ones. To fill your spare time you could research dog sled rides or check out the World Ice Art Championship. Fairbanks has a big movie theater for those cloudy nights.