Best way from Kyoto to Kamakura with JR Pass

Would like to travel with a JR pass from Kyoto to Kamakura going on to Tokyo in the evening. Which is the best way from Kyoto to Kamakura?

Thank you

The fastest is to take JR trains via Ofuna to Odawara and then the direct Hikari to Kyoto. But there is only one connection every two hours.

Next fastest is to take JR trains via Yokohama to Shin-Yokohama and then the shinkansen to Kyoto.

Somewhat slower but easiest is to take the train to Shinagawa and then the shinkansen to Kyoto because only one easy transfer is required.

You can look up the various connections in Hyperdia by de-selecting the NOZOMI option under "Search Details":
http://www.hyperdia.com/

Uji is explaining the reverse: from Kamakura to Kyoto.

According to current timetables, if you take a good connection, the first and the second routes result in arrival at the same time, and the third route (via Shinagawa) results in a slightly later arrival.
If you give priority to convenience over arriving a little earlier, the third route (via Shinagawa) sounds best.
(1)
Kyoto (08:33) ---- Odawara ---- Ofuna ---- (11:43 on weekdays / 11:44 on other days) Kamakura.
Kamakura (12:00) --[Enoden]-- (12:05) Hase.
: Shinkansen Hikari trains to Odawara are much less frequently available than to Shin-Yokohama and Shinagawa.
(2)
Kyoto (08:33) ---- Shin-Yokohama ---- Yokohama ---- (11:43 on weekdays / 11:44 on other days) Kamakura.
Kamakura (12:00) --[Enoden]-- (12:05) Hase.
(3)
Kyoto (08:33) ---- Shinagawa ---- (12:04 on weekdays / 12:03 on other days) Kamakura.
Kamakura (12:12) --[Enoden]-- (12:17) Hase.

A Japan Rail Pass is not valid on Enoden [Enoshima Electric Railway] line.
An ICOCA Card (issued by JR-West) is acceptable on Enoden Line.
I recommend you exit JR via West Gate by showing your JRP then enter Enoden via a nearby automatic gate by touching your IC card to the device.
The one-way Enoden fare from Kamakura to Hase is 190 yen per adult.
http://www.jreast.co.jp/e/stations/e476.html
(A Japan Rail Pass cannot be inspected by a machine, while entry with an IC card needs to be recorded by a machine. So, even if you choose the JR-Enoden transfer gate, you cannot exit JR and enter Enoden via one automatic gate.)

#2, (above) requires 2 changes, at Shin-Yokohama and again at Yokohama.
#3, (above) is just 1 change at Shinagawa.
The Shinkansen tracks and the Yokosuka line tracks are very close to each other.

I just did several weeks ago, The Kyoto-Shinagawa-Kamakura route is the easiest and the time difference is not much.