Chatel is one of the easiest serious skiing destinations drive to from the UK. From Calais it's approximately 525 miles, and normally takes about 9 hours.
Living near London, we've always done the trip in a day. If you live much further away there a lots of great places for an overnight stop.
The latter part of the journey is very picturesque and can be enjoyed as part of your holiday. What's more, if there are more than 2 people travelling it usually works out cheaper, plus you have the benefit of not being restricted on luggage.
We'd suggest you go by the Google Maps route, except we prefer to go via Besancon and Pontarlier after Dole.
If you use this route (or any route including Swiss motorways) you will need to buy a "Vignette" which is the Swiss motorway tax sticker for your car windscreen. You can buy it for about 40 CHF (2014) at the border (you can pay by card or in Euros), post office or some petrol stations.