Continuity mistake: In the first scuffle between Marty and Biff in the school cafeteria, they both grab a hold of each other's shirts until they were stopped by Strickland, but in the next shot, Marty is not holding on to Biff's shirt. (01:40:40)
Continuity mistake: Enchantment under the sea dance: In addition to the comment about Strickland's hands covering his ears twice note some of the people around him have either moved or disappeared. (01:25:55)
Continuity mistake: After Marty borrows the improvised skateboard from the boy, in the first close-up of his feet his tied looped laces are extremely short, but when he climbs over Biff's moving car and then lands on the skateboard, the length of his tied looped laces is significantly longer. It shortens again thereafter. (01:06:05)
Continuity mistake: Moments before Doc and Marty witness the DeLorean's first trip through time, Doc's left hand changes its position on the radio control unit between shots. Initially he's holding a toggle switch, but after saying, "You're gonna see some serious s***," he has a grip on the unit's side instead. (00:21:05)
Continuity mistake: Inconsistent mileage progression on the DeLorean at the beginning of the movie has already been noted. However, mileage when Marty is going back to the future is also inconsistent, both in those scenes and with the beginning of the movie. As Marty is driving toward the cable, the mileage is shown as 33051. Just as Doc begins to slide down to reattach the cable, the odometer is visible again and shows 33044. These scenes were obviously filmed out of sequence, and were filmed before the beginning scenes (32994, 32064, etc) of the movie. (01:39:55)
Continuity mistake: When Marty is running through the car park you can see a person in the background in one of the shots but they disappear in the next shot, then re-appear again.
Continuity mistake: When the clock tower tolls at 10:00, Marty rushes to the DeLorean and as he drives off there are two huge cable spindles at the curb's red edge, near the cloth (lying on the street) that covered the DeLorean. In the next overhead shot, with Doc Brown on the ledge, the cable spindles are gone down below. (01:35:50)
Continuity mistake: When Marty puts the letter in Doc Brown's jacket pocket he puts it in the right one (from the way the pocket is slanted, plus we see the buttons right next to it) but when Doc pulls it out it is in the left one. (01:29:40)
Continuity mistake: Just as Marty is going to be set back to 1985, the camera keeps showing shots of Marty, and the lightning poles coming closer and closer to him. Now Marty is supposed to be going 88MPH and mostly every shot just as the lightning is about to strike is basically the same distance from the lighting poles to the car.
Continuity mistake: At the clock tower, after the tree limb falls, Doc Brown grabs the loose, messy bundle of rope and instructs Marty, but in the next shot when Doc runs up the stairs, the rope is tied up in a rather tidy bundle. This rope bundle changes yet again as he rushes up and then reaches the top. (01:34:10)
Continuity mistake: School Cafeteria 1955, Biff groping Lorraine Baines. Between Marty trying to convince George McFly that Lorraine fancies him and pulling Biff off Lorraine, note the dis/reappearance of the child in a blue/red t-shirt and a man in a blue shirt/tank top sat at the table behind George. (00:57:25 - 00:58:45)
Continuity mistake: Right when the movie begins, a machine opens a dog food can and then pours it in the dog's bowl. When pouring the food, the can isn't moved around at all, but after the food drops, the label is missing a piece. (00:02:25)
Continuity mistake: After the clock tower's ledge breaks and Doc's hanging off the edge, we see a shot from below, then of Doc looking straight up at the clock, which changes to 10:03. But look at the clock in the below shot: the minute hand ticks over behind him, then moves again in the close-up. (01:34:00)
Continuity mistake: When Martly finishes his rambunctious guitar solo at the dance the camera pans the crowd. In the distance on the left we see Mr Strickland uncovering his ears. After this they show a close up of him uncovering his ears again. (01:25:55)
Continuity mistake: In 1955, when Doc Brown pulls Marty's letter out of his coat pocket, he either holds the envelope at its short side edge or at its long top edge, depending on the angle of the shots. (01:32:20)
Continuity mistake: When the JVC recorder is first played, the screen on the TV is a blue colour. Later, when Doc is playing the Libyan part again and again, the screen seems to be a brownish colour. (00:50:10 - 01:05:40)
Continuity mistake: After the scene where Marty and his band are rejected from the school tryouts, Jennifer and Marty are sitting down and a lady comes by and asks them for change to save the clock tower. After they attempt a kiss, Jennifer's dad calls for her from the car. In the background, an African-American man is standing looking in the window of a building, and then walking right. In the next shot, he has moved all the way to the left and is walking right. (00:10:30)
Continuity mistake: When Marty is filming the Doc in the parking lot, Doc says that he needs '1.21 gigawatts' to power the time machine. Marty then says 'Doc, you don't just walk into a store and buy plutonium.'. However, when Marty and the Doc of 1955 are listening to it on his TV, the Doc keeps talking after he says '...of electricity I need'. (00:26:35 - 00:53:05)
Continuity mistake: When Marty is using the phone to call Doc while he is in the diner, his watch (which is on his left arm) starts beeping and the guy behind the counter gives him a strange look. Marty rips out the phone book page and asks the guy behind the counter where XXX Riverside Drive is. Marty is no longer wearing the watch which he just had on seconds earlier. (00:37:55)
Continuity mistake: In 1955, when George walks into his house, Marty thinks of Mystery Science Theatre and looks at the vanishing photograph, and at the dance, after George punches Biff and rescues Lorraine, again Marty looks at the vanishing photo. In these two close-ups the thumb holding the photo looks nothing like Marty's thumb/nail in other close-ups, such as when he powers up the amp at Doc Brown's at the start. (00:03:15 - 01:23:45)
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