(Image from 1966batfan.com)
As you can see from above the Bat-Climb is simply Batman & Robin walking on a fake set with the camera pointed to the side. They attached wires to the capes and pulled on them so the capes would rise to look like gravity is affecting them! As I recall in the third series the Bat-Climb was only used once in episode #96 where Batman & Robin climb upto Barbara Gordon's appartment! The climb was mainly used in the second series, when everyone wanted to be on the show and were queing up to be in Bat-Climb!
Jerry Lewis was the first star included in the Bat-Climb. Series 1 #29 "The Bookworm Turns".
The Green Hornet & Kato appeared in the Bat-Climb, #41 "The Spell Of Tut"
Next they appeared in #85 & #86 to aid Batman & Robin in crimefighting!
Howard Duff was not only in the bat-climb but played Cabala in #119 - "The Entrancing Dr Cassandra" in the third series!
Appeared as Lurch, his role in the Addams Family. I believe this Bat-Climb was from episode #61 "The Penguin's Nest"