The White Mountain road is usually closed during the winter which, as WBTravis has said, makes for a long ski or snowshoe to White Mountain Peak.
The Mount Barcroft Station provides tons of weather data:
http://www.wrcc.dri.edu/weather/barc.htmlThe Winter weather in the White Mountains can be quite radical. The highest wind recorded at White Mountain Peak was 162.4 MPH in December, 1968. Temperatures can reach well below -25 F.