To use shinkansen is fastest from Tokyo to Kyoto. But you can take some trains of Tokaidosen with some lines, you need to change the short-haul routes trains around 7 times. It will take around 7 to 8 hours. If you will buy the thicket the name of "青春18きっぷ " , You can save the money ￥8,040.
The following route is faster than you wrote (from Tokyo station to Kyoto). To take a train from Hamamatsucho station to Shinagawa station: 6 min by JR Yamanote line or Keihin Tohoku line From Shinagawa station to Kyoto: 2 hrs and 5 min by Shinkansen (subcategory Nozomi)