I've been into Katahdin 5 times in winter, but not since 1991, so a few things may have changed since then. However, you will need to write to Baxter State Park at 64 Balsam Drive, Millinocket, Me 04462 to get an application to go in winter, and they require a minimum group of 4 people, each filling out a climbing resume.
The first day involves skiing 12 miles from Togue Pond into Roaring Brook bunkhouses. Day two involves hiking or skiing with skins the strenuous but beautiful 3.3 miles to Chimney Pond. You should arrive with enough time to do a climb on the Pamola Cliffs if you have any energy left.
If you just want to summit, the Cathedral Trail is your best bet, but I highly recommend the Dudley Trail which leads to the Knife-edge and eventually the main summit with a descent via the Cathedral Trail. There is nothing as cool in the North East as the Knife-edge in winter, but it is very exposed, often technical, and you may want a rope.
If you want to go for a route, Cilley Barber is a serious undertaking, but spectacular. Near the top, when the climbing eases off, climb left onto the ridge for added excitement and follow it to the main ridge.
Also good are Gallery Route, which involves a bit of rock work in normal conditions, or either of the Pamola's Fury routes. Of course you can go for steep, fat flows on the Pamola ice cliffs; Hands Across the Water is a good one.
Keep in mind that once you are at Chimney Pond, the ranger can prevent you from climbing if the weather is bad ...
Hope this helps; it's a great place to be!