I have been visiting the surrounding areas of Mt. Fuji for the last ten years and would have to say the easiest way to get around quickly is in a hire car. Most of the best viewing points are from the lakes and hilltops. The distances to each lake with public transport are far and not frequently available.
Most tourists take a day trip from Tokyo, but I would suggest an overnight stay. This will give you a chance to catch a sunset and sunrise. Mountain weather can be very unpredictable so be prepared to have heavy cloud cover on