I hate spending tons of money on equipment for photographing the Northern Lights / Aurora but it is also frustrating when shooting the lights with sub-par equipment. To take the best shots you really need a full frame DSLR camera but even for a great used one with a used lens, you are looking at over $1000 so here is the best sub $1000 option on the market right now.
This camera has a cropped sensor, that is, it is the second best sensor size to have, only falling behind the full frame sensors. This camera has fully manual mode, a must for Aurora shooting, plenty of mega pixels with 24.2, and ISO levels from 100-25,600. It also has the new EXPEED 4 image processor which helps deduce noise and allow for cleaner high ISO shots (important for shooting the Northern Lights).
One of the huge advantages of getting this camera is that lens can be bought that are also compatible with Nikon full frame cameras so when you decide to upgrade you can take your lens with you. Speaking of which, you can buy this camera with the kit lens, and that lens should serve you well for day to day use for about $500 here but check the other sellers because as of writing this you can get the body and kit lens refurbished for under $400, with a variety of used Nikon D3400s for around that price point.
Now here is where things get tricky, you need to get a lens that is compatible with the D3400 but if you want an easy upgrade path, you need one that is also compatible with Nikon's full frames. I would suggest starting with something like the Rokinon 14mm F/2.8 lens. It does several things right, first, it is a wide angle lends at 14mm so you can capture a large portion of the night sky. Next, its aperture gets as large as F/2.8, which is large enough to let in enough light for good Aurora and astrophotography shots. You should be able to pick one up for under $400 and a used/ open box one for about $300.
Don't forget extra batteries, a nice case to protect your investment, and a sturdy tripod, preferable with something that is not metal to hold onto because you can frost bite your hands quickly if you grab on to -40F metal!
Before you decide what to bring on your trip to Alaska for Aurora viewing consider what time of year you will be visiting. The chart on this page is meant to give you a good idea of what the average weather is like. Pay attention to the average lows as they tend to occur at night, this is the time you will be out viewing Aurora. Soon before you depart for your trip you should check the long term weather forecast for a better prediction of what the weather will be like when you are here.
Base Layer: Feet
Having a base layer of insulation all across your body will greatly aid in keeping you warm and comfortable while out viewing Aurora. I would highly recommend wearing wool or a wool substitute sock.
Base Layer: Legs
Moving to your legs, there are two strategies that I find useful, either wear insulated or flannel lined pants or long underwear under the pants you already own. I personally prefer the insulated pants option because I dislike the feeling of long underwear and feel like it doesn't let my legs breath but if I lived in a climate where those pants would have little value I would just get long underwear to save money.
Base Layer: Body
I find wearing a long sleeve moister wicking shirt is the warmed base layer for the chest. Most of us have heard of Under Armour and their shirts work great but something like this Hanes shirt works just as well and will save you money. It is important that this layer is not cotton.
Next you should have a warm shirt like a flannel shirt or a wool shirt. Now that you have a nice base layer and a warm shirt you should wear a nice warm middle layer like a fleece jacket or this is what I personally wear.
Base Layer: Hands
Don't forget to bring some light gloves with you at a minimum. Some people like to wear light enough gloves that they can still operate their camera or use a smartphone app to control the camera. Regardless, this is one of the most difficult areas to dress for because of freezing fingers in cold temperatures and the need to use the camera.
Base Layer: Head
For temperatures closer to freezing you can get away with only one layer for your head. Something like this trapper hat will keep your head warm and help keep your neck and ears warm too. For even colder temperatures get something like this turtle fur that will add insulation to your head and provide a protective layer all around your neck.