In reply to Mel:
> Does anyone know what is the best way to get to the mountain in May/June, and from which station to start walking
Depends where you are coming from, but if it's the Tokyo area take the train to Fujisan or Kawaguchiko station, then catch a bus up the Subaru toll road to the Yoshida trial 5th station.
> Is it necessary to use huts, and will any be open?
No, and most likely no for May/June.
> Also, which routes are best to take at this time of the year (depending on conditions)? Am I being daft attempting it (as some websites say), or is it feasible to do it? I am an experienced mountaineer, climb Alpine AD and Scottish IV, and I will be bringing my axe and crampons.
Go for it. The routes aren't technical at all, just be sensible and take care with weather (e.g. cold, wind, sun), altitude, snow conditions, etc. I've not done it outside the official climbing season, so this is my best guess for what the hazards would be around that period!