Continuity mistake: When Marty rides to the mall he is riding his skateboard, but after Doc comes out in the car, it's gone, then at the end of the film when Doc drops off Marty he has it again. (00:18:40)
Continuity mistake: When the DeLorean sets off for the lightning strike, you can see it leaving tracks and spray on a wet road, later there is a close up of one of the wheels, which shows the road to be dry. Throughout this scene the road appears to alternate between wet and dry, during what is supposed to be a storm. (01:39:00)
Continuity mistake: Lorraine Baines' hands are on her hips, a brief cut to Marty and back and she now holding a cup and saucer at chest height. You could argue she had enough time to pick it back up, but this would seem odd as she's only just put it on the table. (01:43:55)
Continuity mistake: At the beginning, when Michael J Fox is cranking up the amplifier, the dial marked "Primary power" initially rises from 100 - 120, then when he turns the "overdrive" knob, it's back at 110 and goes to max. (00:03:15)
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)
Visible crew/equipment: A boom mic pole is reflected in the glass near Doc after Lorraine and her friends run past him in the school hallway. (00:57:40)
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)
Revealing mistake: When Marty is skateboarding and being chased by Biff's car, Marty's stunt double is noticeable because his wig is bulkier and darker coloured than Marty's real hair. He looks just like the Trololo guy from Youtube. (01:40:50 - 01:41:25)
Other mistake: When the Libyans shoot at Doc Brown and the bullets hit his van they leave no bullet holes, just impact sparks. (00:29:10)
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)
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