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: Some of the items in Doc's lab change position between the shots of Marty standing in front of the amplifier and it exploding. (00:04:15)
Continuity mistake: When Marty looks at the car he is about to crash into, there is no one standing on the pavement, but when he jumps onto it, a man appears. There was not enough time for him to appear. (01:07:00)
Continuity mistake: When Doc takes the photo from Marty, the way he holds it changes between shots. (00:50:07)
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: 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: When Marty arrives at 1955-Lyon Estates he removes his hood and his fringe is brushed from right to left. When the angle changes it has swapped sides.
Continuity mistake: Right before Doc presses the remote control to send Einstein to the future, his left index and thumb fingers are on the remote. In the close-up it's just his thumb on it. In further close-ups it swaps to the index and thumb.
Continuity mistake: Before Doc sends Einstein to the future and says "If my calculations are correct...", Marty raises his left arm. When the angle changes it's lowered.
Continuity mistake: While Marty is talking with Jennifer in the square he is holding a tape case in his left hand which disappears in the next shots.
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