Visible crew/equipment: When the clock tower ledge breaks, Doc Brown slips and loses his grip on the electric cable. In the next shot the stunt wire pulling the thick electric cable towards Doc's pants is visible. (01:37:50)
Visible crew/equipment: When Doc Brown loses his footing on the clock tower as it begins to toll, in the next shot, when he swings about towards the clock the thin black stunt cable attached to his waist, that leads around the neck of the statue, is visible. Do not confuse it with the thick silver electric cable Doc is holding. (01:35:25)
Visible crew/equipment: In the scene in the beginning where Marty meets up with Doc for the first time, Doc shows Marty how the DeLorean works with the remote control with Einstein in the car. In the shot that switches to see Doc put the car in reverse on the remote control, you can clearly see a crew member manually steering the car in a dog costume (You can even see his hands turning the steering wheel). (00:20:30)
Visible crew/equipment: When Marty is riding past the cafe on his skateboard and all the chicks are waving at him, you can see the reflection of the camera lorry and crew behind him. (00:06:25)
Visible crew/equipment: A boom mic pole is reflected in the glass near Doc after Lorraine and her friends run past him in the school hallway. (00:57:40)
Visible crew/equipment: At the end of BTTF they all pile into the DeLorean which reverses up to the camera where the matte box and what looks like the top of the cinematographer's head are reflected in the chrome number plate. (01:46:50)
Visible crew/equipment: When Marty arrives at the 1950's town and is about to be run over by a green car, the whole filming crew is reflected on its door. (00:35:40)
Visible crew/equipment: In the opening scene, when the TV turns on as the news comes up, you can see a man with a white shirt in the reflection of the TV. (00:01:50)
Visible crew/equipment: A boom mic reflection is visible on the vent window of Doc's van when Marty asks him if the plutonium was ripped off. (00:26:35)
Visible crew/equipment: When Marty asks if the time machine is nuclear, a boom microphone is visible in the right windows of Doc's van. (00:26:28)
Answer: The video camera was in the DeLorean. With the right kind of adapter, which was common enough in the 80s that Doc might've had it on the camera or been able to jury-rig something in the 50s, it would have been possible to connect it into the antenna screws in the back of the TV like an old Atari and play it directly from the camera.
Captain Defenestrator
TVs in the 50s had a two prong antennae connection (two screws in the back that you put a prong antennae into) TVs in the mid 80s also had this. The coax connection (the one wire that screws in) was starting to become common, but, the two prong connection would have been more likely on any given TV at the time, so, whatever wire they used to preview recordings probably had that. very convenient that Marty brought those cords with him.
An old Atari 2600 RF Adapter would be how one would link a video camera to an old-fashioned television. A simple-enough part that Doc could probably make one with 1950s technology.
Captain Defenestrator