Early Winter is a great time to see Mt. Fuji - the crisp air and clear skies often result in very nice views of Fuji. If all you want to do is have a good look, two great places are Kawaguchiko (Lake Kawaguchi) as well as Hakone, which has two (of many) great spots in Ashinoko (Lake Ashi) and Owakudani.
You can get to Kawaguchiko Station by taking the Shinkansen to Mishima, and then transferring to a local line. It takes roughly 2 hours. Getting to Hakone is far easier (and quicker); you take the Shinkansen to Odawara Station, then take the local line to Hakone-Yumoto Station. From there, you can either bus it or take a taxi to Owakudani or Ashinoko. Or just rent a car - it gives you the most flexibility and it's easy to drive in most parts of the country.
Before you go, make sure to check out one of the various forecasts as well as webcams to check that Fuji will be visible. Early morning (and to a slightly lesser extent, late afternoon) will give you the best shot at a clear view.