Revealing mistake: 1955 skateboard chase scene around town square. Note in this scene a metal device has been added between the kicktail and the back truck of the skateboard to create the sparks, then disappears and reappears throughout the whole chase scene. (01:03:40)
Continuity mistake: When Marty is holding on to the back of the truck, Biff attempts to hit him with his car. Marty quickly pulls himself to the side of the truck to avoid being hit but in the next shot, he is holding on to the back of the truck.
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