Hello there,
There is not JR station near Kawaguchiko. You have two options going to Kyoto.
1.) Is to travel to Otsuki station using the Fujikyu railway 1.140/yen one way. From there you can use local lines to either Tokyo or Shin-Yokohama stations and there use the Shinkansen to Kyoto station.
2.) The other is to use an express bus from Kawaguchiko to Mishima station and from there use the Shinkansen to Kyoto. Bus details are here: http://bus-en.fujikyu.co.jp/highway/detail/id/29
Option 1 will cost a little less, as the JR Pass is valid from Otsuki station. Option 2 is more convenient, as there are fewer transfers but the bus has a higher fare of 2,260. Total travel time depends on what time of day you go and whether.
Hope this helps,
Daniel