Continuity mistake: While being chased by the terrorists Marty passes the J C Penny's store, drives a little farther and passes it again. Just before the Libyans aim their rocket at Marty, their van changes from about 100 yards from the store to right in front of it in less than a second. (00:30:25)
Continuity mistake: When Marty is standing over his father's bed in 1955, Marty puts the headphones on George and then puts the tape in the cassette player. If you notice the hairdryer is on the left side of the belt and is touching his left arm, but when they cut to Marty holding the tape player out in front of him the hairdryer is on the right side. (01:02:10)
Continuity mistake: In one shot of Einstein strapped in the DeLorean, if you look closely in between the steering wheel, the yellow time circuits on the bottom read the month as "10" rather than "Oct." And the red and green time circuits read "11" instead of "Nov." (00:19:20)
Revealing mistake: Right before the white smoke shoots out of the time machine after Einstein arrives one minute in the future in the beginning of the film, you can see the 2 fire extinguishers built in to shoot out the "smoke". (00:23:40)
Trivia: The amplifier that Michael J Fox blows up at the beginning is labeled as being a CRM-114. This is a reference to Dr. Strangelove, in which the B-52 crews receive their orders over a CRM-114. It was also the serial number of the Jupiter explorer in 2001: A Space Odyssey (1968) also directed by Stanley Kubrick. (00:03:05)
Revealing mistake: When the DeLorean is struck by lightning near the end of the film, if you look closely, you can see that there are two people in the car, when it should just be Marty. You may have to look closely, but you don't need to zoom in to see it. Also, since they never re-shot this scene, you can see it in all three Back to the Future films. (01:40:40)
Continuity mistake: Beginning of film. When Marty arrives at Twin Pines as the Doc requested he cruises up alongside Einstein on his skateboard, saying "Where's the doc, boy?" There's a cut to the truck opening, then back to Marty, and for some strange reason the yellow wheels on his skateboard have turned pink, but the reflection on the wet car park is still yellow. It cuts back to the DeLorean coming out of the truck then back to Marty, and his wheels are yellow again. This doesn't happen in the VHS version. (00:17:40)
Continuity mistake: When George goes into the cafe to tell Lorraine he is her density/destiny, the people sitting at the bar keep changing with different shots. Best example is just after George refers to his "density" - in the background there's a brunette in a blue jumper at the bar. A blonde woman and a short-haired guy then sit on her left, but then disappear in the shot of Marty sitting at the bar - on the brunette's left there's now another brunette in a pink jumper with a different guy. The brunette's got shorter hair too. In the very next shot the blonde's back with no sign of Marty. (01:02:10)
Continuity mistake: At the Twin Pines mall during the chase scene with the Libyans, there is one shot where you get a great look at the driver as he's trying to start the engine. Later on you get another look at the driver as the van is moving and it's a totally different person. (00:29:52)
Continuity mistake: After Marty has been thrown into the bookshelf after the amp explodes, he is lying on the floor with his left leg bent at the knee. In the very next shot however, he has not moved but both legs are straightened and on top of each other. (00:04:05)
Revealing mistake: When Marty is being chased by the terrorists, when he makes the 2nd turn, if you look closely at Marty it is obvious that it is not Michael J. Fox behind the DeLorean windshield, rather it's a stunt driver resembling the original Marty, Eric Stoltz, who was replaced early in filming. (00:30:45)
Continuity mistake: 1955 Skateboard Chase Scene around Town square. After Marty falls over the couple, a shot of Biff's car shows all the passengers standing up, but in the very next frame everyone is sat down. Also, that is not Biff driving. (01:04:20)
Continuity mistake: At the beginning of the movie after Biff leaves the McFly home. There is a scene where Marty and George are talking. In one shot George has three pens in his pocket, a few seconds later he has two. (00:14:14)
Continuity mistake: While Doc is waiting for Marty to get back from the dance, he looks at the watch on his left hand, his pocket watch, and the watch on his right hand. When he looks at the watch on his right hand, he is wearing it on the inside of his wrist. Then, as he is running around the car, he has no watch on his right wrist at all. Moments later, as he is pointing down the street, it is on the outside of his right wrist. When they are checking the alarm clock, he holds it in front of the alarm clock to check the time, and the watch is still on the outside of his right wrist. (01:31:15 - 01:32:30)
Revealing mistake: When Marty is being chased on the skateboard in 1955 he skids rounds a corner and you can see that this scene had been shot before as there are skid marks there from previous takes. (01:03:20)
Continuity mistake: When Marty arrives in 1955, he lands on Peabody's farm and rams through the pine tree he then hits a scarecrow which flies over the top of his car. Then as they show it from the inside the scarecrow slams into the windshield - but how can it slam into his windshield if it is already over the car? (00:31:30)
Continuity mistake: When Biff's car slams into the manure truck, in the first shot the side of the truck is adjacent to the car's back passenger door, but in the next overhead shot the side of the truck is adjacent to the front windshield. (01:07:20)
Continuity mistake: When Marty's been thrown back by the amp, the papers on top of him change between the first and second shot. Precisely a Polaroid photo, a poster/map and a vase are missing. (00:04:20)
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)
Revealing mistake: At the end, the DeLorean hits the left dustbin and comes to a halt - you can easily see that it's not Christopher Lloyd driving, it's someone with a much bigger build, wearing an awful white wig. Further, he isn't wearing sunglasses until he gets out of the car. (01:45:55)
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