Tokyo weather in June

Is Tokyo hot in June? I found from this website, and saying it's max. is 25C. Is this correct? Does it start rain too? I have one more concern. Are we still able to see snow on top of the Mt. Fuji?

Is Tokyo hot in June?

Some days can be hot and humid, others can be still kind of chilly.

Does it start rain too?

Yes, June is the rainy season:
http://www.japan-guide.com/e/e2277.html

Are we still able to see snow on top of the Mt. Fuji?

In the average year, the snow on top of Mount Fuji is disappearing during the month of June. So, usually, you can still see some snow in early June, bu no more snow towards the end of June.

Even though the rainy season starts in June (I would have said mid-June, not the beginning), june is one of the nicer months weather-wise. It's not like a monsoon where you get torrential downpours, it's more drizzle and overcast interspersed with fine days.

The temperature is usually quite pleasant, although it can be a bit cool in the mornings some days in early June.

Even if there is snow on Mt. Fuji, the visibility may not be good enough for you to see it, even nearby- from about late April it gets covered in cloud most days. You might be lucky though.