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.
Continuity mistake: While Doc is explaining to Marty how the machine works, he has a device on his index finger on his right hand that moves in the next shots.
Continuity mistake: When Biff and his friends are chasing Marty on the skateboard, just before they hit the manure truck there is canvas covering the fertilizer which disappears in the next shot. (01:07:45)
Continuity mistake: After Marty wipes out in front of the town hall, he clutches the hood ornament on Biff's Ford. As the car turns, Marty moves himself towards the fender, but between shots he repositions and is now clutching the hood ornament like before. (01:07:05)
Continuity mistake: When Marty is in 1955 observing young Uncle Joey in his playpen, the colored balls on the second rung keep changing their positions from one angle to the next. (00:45:05)
Continuity mistake: When Marty is on a skateboard being chased around the town square by Biff and his friends, he grabs the back end of a truck. After the truck makes a right turn, notice how the two kids he initially stopped to get a skateboard from at the beginning of the scene suddenly appear standing at the street corner between shots, holding their wooden crates. (01:06:25)
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.
Continuity mistake: When Marty drives out of the Peabody barn after arriving in 1955, it's night, or certainly pre-dawn - the sky is very dark from every angle when he's fired at while driving away. Next shot he drives past the "Peabody ranch" sign onto the main road and it's significantly brighter, with the sun rising over the horizon. Then when he notices the Lyon Estates sign shortly afterwards it's broad daylight - judging from the length of the shadows the sun's very high in the sky. Lyon Estates isn't that far from Twin/Lone Pine(s) mall.
Continuity mistake: When Marty is on a skateboard being chased around the town square by Biff and his friends, he grabs the back end of a truck. He does with two hands at the same point, then uses one hand in the next shot, then two hands again.
Continuity mistake: As Marty watches George pedal away from the café, the amount of sunlight cast on the town square decreases greatly between shots. (00:41:00)
Continuity mistake: As Lorraine's family watches "The Honeymooners" around the dinner table, a round pillow on the sofa nearest the lamp repositions itself. (00:46:05 - 00:47:30)
Character mistake: In the parking lot when Marty begins to film, Doc says the date and exact time according to his watch. Marty then looks at his watch. Apparently his is off because he shakes his wrist and puts the watch to his ear to see if it is still working. However, he is wearing a digital watch which wouldn't tick or likely have broken parts to rattle.
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)
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