Mount Fuji is one of the greatest and well known icons of Japan. I personally have what has been called an enormous obsession with this grand mountain. Although I have been to the Mount Fuji area on three different trips I have yet to see, during day light hours, Mount Fuji not being completely obscured by clouds (It is my mission to be so this year). That being said many people find it difficult to get to the region as, thanks to the mountains in the area, it can be tricky to access. I will go through a couple of the main routes to get to see Mount Fuji from the Kawaguchi-ko area.
The highway bus is one of the easiest way of accessing the Mount Fuji area from Tokyo, with no need to worry about transfers of getting on the wrong route. The ticket costs 1,750 JPY to travel from the Shinjuku Expressway Bus Terminal to either Kawaguchi-ko station, Fuji-Q Highland or, at certain times of year Mount Fuji 5th Station and the Fuji Shibazakura Festival. I would recommend booking in advance and reserve your seat. See link below;
From Shinjuku station in Tokyo take the LTD. Express Kaiji 107 to Otsuki station, the journey takes around 60 minutes. From Otsuki station take the Fujikyu Railway train to Kawaguchiko station, all in all the trip will cost around 2000 JPY. I enjoy the Fujikyu railway as the trains are decorated in such a cute way!
Many people wish to take the Shinkansen, especially if they have a JR pass, however the Shinkansen route is not the best to get to somewhere like Kawaguchi-ko station. You can however get to Hakone by using your JR pass up to Odawara station, then you can use the Hakone Free Pass to get further. You can, on a clear day, see Mount Fuji from here.
If you would rather be whisked away then there are a number of guided tours which go to the Mount Fuji and Hakone region. I have been on such tours but really prefer travelling without a tour group, but it can be a less stressful way to get around (although much more expensive).