Continuity mistake: After Marty exits the cafe to make himself a skateboard out of two wooden crates, the crates are half a meter apart from each other, but close in the next angle. Also, the chubby kid holding his friend's crate swaps to holding his between one frame and the other.
Continuity mistake: When Marty gets hit by the car and wakes up in Lorraine's bedroom. She constantly calls him Calvin. Marty says "Calvin, why do you keep calling me Calvin"? When he says this he puts his arm on his lap. In the next shot behind him when Lorraine replies, his hand/arm is by his side. (00:43:40)
Revealing mistake: When the DeLorean travels back in time after Marty returns to 1985, the light emitting from the DeLorean does not actually come from the DeLorean, but from the side of the screen instead. (01:44:16)
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)
Continuity mistake: When George is in the school cafeteria talking to Marty about science fiction stories, he holds the notepad differently with each changing shot. (00:58:55)
Continuity mistake: While Marty's 1950s family is having dinner, the milk bottle's level is one mark below the red printed label. A second later it's 5 centimeters above the red mark.
Continuity mistake: When Doc puts Einstein in the car, the dog turns his head inside the car. A frame later, from a new angle, it's outside the car.
Continuity mistake: When Marty was recording, Doc never hesitated between "reaction" and "to", then when they are watching the tape, he hesitates a little.
Continuity mistake: When Goldy (or Golby?) says, "Mayooorr", George is facing Marty. Then when Goldy starts saying, "Now that's a good idea," George is facing Goldy, without any motion of turning. (00:40:35)
Continuity mistake: When Doc and Marty are watching the DeLorean burn out from across the vacant Twin Pines Mall parking lot, the radio control unit's antenna is seen lowered one shot then fully raised in the next, despite Doc not having touched it. (00:21:20)
Continuity mistake: When Marty sees the Lyon Estate works there's a mild breeze blowing. Just a frame later, from a new angle, there's a very strong wind. (00:34:18)
Continuity mistake: The time machine is gone for 1 minute and 21 seconds - not just 1 minute. (00:22:00 - 00:23:20)
Continuity mistake: When Doc takes the photo from Marty, the way he holds it changes between shots. (00:50:07)
Revealing mistake: When Marty begins to slump down due to fading from existence look at the guy playing the piano behind Marty, he's not even pressing the keys on the piano just putting his palms on them repeatedly.
Continuity mistake: George's hair fixes itself when he tells Marty about Darth Vader coming down to visit him. It's initially messier.
Continuity mistake: When Doc and Marty are looking for his dad at school, you can see a girl with a blue sweater going downstairs twice. (00:56:49)
Continuity mistake: When Marty stops the Delorean in front of the Lyon Estates sign after escaping the farm, notice the sun - it's overcast and kind of foggy. When he looks at the sign from another angle, it has become more sunny, and later in the day. When the couple with the car passes, it has returned to the same as before.
Continuity mistake: When Marty confronts George as "Darth Vader", after George asks "who are you?" Marty plays the music again, and George presses his hands to his ears. In the shot from behind, his hands are cupped around his ears, in a very different position. Then when Marty says "silence earthling", in the shot from the front his hand is away from his ear entirely. (01:02:35)
Continuity mistake: After George knocks Biff behind Marty there is a couple running to George, but in the open view they disappear.
Continuity mistake: When George is checking the time at the prom there is a girl in grey behind him that changes to one in pink. (01:18:56)
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