Skiing/Riding Mt Washington can cover a whole range of trips. The standard trip for the ages is the hike up to Tuckerman Ravine in spring and ski/ride the many routes there. While you won't be alone up there, it's still backcountry. You're responsible for taking care of yourself with food, water, gear, etc. There is avalanche, ice fall, and terrain dangers that are not trivial. Trying to ski there earlier in the winter season is much riskier, with the avalanche danger in particular ranging from dangerous to severely dangerous.
There are many other places to ski on the Mt Washington massif. Other ravines (Gulf of Slide, Oakes, Great Gulf) offer runs of all levels of difficulties. They are all much more remote and shouldn't be taken lightly. Those are not places to try without someone who has been there before and can evaluate avalanche and terrain dangers.
A fine trip in mid-winter is to skin up the Tuckerman Ravine trail to HoJos or the base of the ravine, and then ski back down the Sherbourne trail. That's a good way to get a feel of the distances and terrain. There are several ski trails around Pinkham, such as the Avalanche Brook trail that offers more touring options.
Get the David Goodman book "Backcountry Ski Adventures (NH and ME)" to start, and find some folks to ski with. I don't recommend heading up there solo if you've never done backcountry skiing before, or if you've never skied on Washington before.
-dave-