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 up to Barbara Gordon’s apartment! The climb was mainly used in the second series, when everyone wanted to be on the show and were queuing up to be in Bat-Climb!
15 people guest starred in the Bat-Climb, one of which appeared in the 1966 Movie. I believe it was George Cisar, the other 14 are listed below. The images are from 1966batfan.com.