Continuity mistake: Before Marty kicks the speaker while playing Johnny B Goode, there is a case of Pepsi below it which disappears between shots.
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)
Continuity mistake: After Marty gets to the square in 1955 his hair changes after he grabs the newspaper.
Continuity mistake: While Marty is at Lou's Cafe, Biff and his gang enter. When Biff starts bullying George, note George's haircut with shaved sides, back (above his neck), and sideburns. In the following shots, George's haircut with its shaved areas change length. This happens again, when George is with Marty and then tries to ask Lorraine to the dance. (00:39:00)
Continuity mistake: Throughout most of the film, the licence plate on Doc's car reads 8N 39742, but after the DeLorean travels back to 1985, it changes to 7F 42857. (01:41:31)
Revealing mistake: As the DeLorean's tyres start igniting as it is about to travel back in time, the flames are not coming from the tyres, but from some ignitors attached to the car. (01:40:36)
Continuity mistake: When Marty approaches the couple, the couple goes down a step twice. (01:07:03)
Continuity mistake: When Marty skates off after knocking over the couple, the car behind him is red and cream, but in the next shot, it turns completely red. A yellow car also appears in the second shot, when it was not present in the first. (01:07:10)
Continuity mistake: At the school when Marty is about to fight Biff, in the background there is food that changes position between shots.
Continuity mistake: After Marty wakes up at his mother's place, she sits on the bed and leaves her coat that changes position between shots.
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)
Deliberate mistake: The sequence with the Libyans chasing Marty at the end of the film is much shorter than the original chase.
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