Hi Sam,
Hokkaido is pretty far away from Tokyo, a one way trip Tokyo - Kutchan by train takes 575 Minutes (almost 10 hours non stop) so many people decide to fly from Tokyo to Hokkaido. It is however hard to say what is more economic as airplane ticket prices change a lot.
If you do decide to take the train than the JR Pass will offer some really nice savings over normal train ticket prices. For instance the normal price of a one ticket Tokyo - Kutchan is ¥ 21,740, the 7 day JR Pass is ¥ 28,300 so it's very easy to make savings with it. I've looked up the route for you, so you can get an idea of what is involved. With the JR Pass you can take the Shinkansen from Tokyo - Aomori for there take a limited express train to Oshamambe and finally a local train to Kutchan.
There's a night train going Ueno - Aomori called the Akebono express which is covered by the JR Pass but the train is rather popular and can sell out quickly, so I think it's better to take the Shinkansen to Aomori. You can find more information about the Akebono train here.
Have a great time enjoying the snow!
Daniel