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)
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)
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: 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.
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