If you would like to see Mount Fuji, the most important thing is checking the WEATHER FORECAST.
Moisture could prevent the sight of it, so also check the humidity. On a rainy day, you could hardly see her.
There are also Live Cameras that you can check how it is online:
The NORTH SIDE of Mount Fuji: Five Lakes area (Yamanashi Prefecture), which will take one hour and a half by bus at least to get to a Shinkansen (Kodama) Station, such as Mishima Station:
The SOUTH SIDE of Mount Fuji: Shizuoka Prefecture, where the bullet trains run:
The EAST SIDE: Hakone (Kanagawa Prefecture):
You need to go up to Oowakudani Valley or Lake Ashi to see it.
It cannot be seen from Hakone Yumoto or Goura Park because of the surrounding mountains.
The posters and pictures you see is basically shot on clear days during Autumn or Winter.
I wish your luck but it is becoming the season that you need tremendous luck to see the highest mountain in Japan. Being flexible with your schedule will raise the possibility of seeing it, but if it is rigid, maybe you can only see it at the museums with its pictures. Good luck!!