Continuity mistake: When Marty rides off after borrowing the skateboard, a woman in the background bends over towards the end of the shot, but in the next shot, she is standing upright. (01:06:10)
Deliberate mistake: The sequence with the Libyans chasing Marty at the end of the film is much shorter than the original chase.
Continuity mistake: In 1955, when Marty shows Doc the flux capacitor, he points the flashlight in different directions between shots (first at Doc, then the flux capacitor). (00:51:35)
Continuity mistake: After Old Man Peabody fires his rifle at Marty, it changes position between shots. Initially the rifle is trained on the barn doors, but when the DeLorean busts out a moment later he's got it aimed at the ground instead. (00:33:05)
Continuity mistake: As Marty takes in his surroundings at the 1955 town square, he pauses for a few seconds to glance at the Roy's Records storefront. In that short span of time, the cloud cover changes entirely. (00:35:55)
Continuity mistake: When Marty pulls the family picture from his shirt pocket in front of George's house, he holds it with both hands at first. But once the angle changes, only his right hand has a grip on the picture. (01:01:45)
Continuity mistake: When George walks to Lorraine with the flirting words written on a notepad, her purse changes positions between one shot and the other.
Revealing mistake: While Biff and the others are chasing Marty on the skateboard, when Marty grabs a green truck the stunt man is doing it for him.
Continuity mistake: When Marty takes a swig from the bottle, we see the motion of Lorraine's hands as she lights a cigarette. She finishes lighting and then grabs it with her left hand. In the next shot where Marty spits alcohol all over the place, Lorraine is busy lighting her cigarette a second time. (01:18:10)
Continuity mistake: As Doc reacts to the icy DeLorean, the door Doc was about to pull open barely has any ice on it, but when Doc opens the door with his foot, more ice is on it, then melts after Einstein runs back into Doc's truck. (00:24:00)
Revealing mistake: As the DeLorean's vanity plate spins around in place, you can see how it's attached to a short steel post, stuck into a hole in the pavement. (00:22:10)
Continuity mistake: When the clock tower bell sounds a third time, Doc backs away from the ledge and stares down below in shock. Doc's arms are hanging down at his sides, but once the shot changes he's seen cupping his ears. (01:35:35)
Continuity mistake: As Doc turns the plutonium jar on the back of the DeLorean, the little thing sticking out of the jar slides greatly down, without any motion of moving. (00:27:15)
Audio problem: Marty's singing during the "Johnny B. Goode" number is obviously dubbed; it goes out of sync several times. (01:27:40)
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