Continuity mistake: In 1955, when Doctor Brown is watching the video of himself, he sees himself saying that the Libyans are coming. You can hear a significant pause between "They found me" and "I don't know how but they found me." But in the beginning of the film he says the entire phrase without any pauses. (00:28:45 - 01:08:10)
Continuity mistake: In the cafeteria, when Mr. Strickland is glaring at Marty and Biff, the table Marty was sitting at with George is behind him. When Biff leaves there is a close-up of the table and both the lunch tray and Marty's jacket have changed positions significantly from the previous shots (George had already left). (01:00:30)
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)
Continuity mistake: As Marty drives the DeLorean towards the clock tower, if you look carefully you can see that he drives past the 'Bluebird Motel' billboard at least 3 times (which is where he started from) and drives past that small building with the slanted grey roof in the square about 4 times.
Continuity mistake: When the Libyan terrorists show up in the VW bus, there is no license plate at the front, but in some of the following shots during the scene a blue license plate is affixed to the front bumper. (00:28:55)
Continuity mistake: After Lorraine drops Uncle Joey's cake on the kitchen table, Dave can be seen resting both hands on the back of his chair. When the shot changes, both of his hands are elsewhere. (00:15:00)
Continuity mistake: After Marty drops a quarter in the clock tower fund can, he briefly rests his right hand on top of its red lid. But at the next angle, his left hand is on the lid instead. (00:10:45)
Continuity mistake: The second time the Libyans kill Doc, after the DeLorean disappears, the kiosk is in flames. A frame later the flames are gone. (01:44:19)
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: When the lightning bolt hits the clock tower in 1955 and sends the DeLorean back to 1985, Doc Brown gets up and looks at the clock tower. What happened to that awful lightning storm of only a few seconds earlier? Not even a trace of wind is present in Doc's hair. (01:36:55)
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: 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 dance, when Marty is parked with Lorraine, Biff reaches into the car and grabs the front of Marty's shirt/tie, but in the next exterior shot, Biff now has got hold of the back of Marty's suit. (01:20:10)
Continuity mistake: When Marty gets out of the DeLorean to look at the "Lyons Estates" markers it is supposed to be a little past sunrise. The shadows should be very long, but all of the shadows are small. The scene must have been shot much closer to noon. (00:32:40)
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)
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: In the scene after Marty returns to 1985 he jumps out of the DeLorean to verify he's back and then jumps back in the car to go save Doc at the mall. In the shot where he's putting the car in gear, you can read part of the yellow "Last Time Departed" display on the time circuits as "1985, 1:20AM", and in the shot where he's trying to start the car, you can read "Oct". It should read "November 12, 1955, 10:04PM" since that was the time he just left. On top of all that, the display appears to relate to when Einstein traveled a minute into the future (you can see when Doc shows Marty the time circuits after he lets Einstein out that the "Last Time Departed" is "October 26, 1985, 1:20AM"), not when Marty traveled back to 1955 (October 26, 1985, 1:35AM). You could argue that the "Last Time Departed" display is not working properly, but Doc uses Marty's last time departed (October 26, 1985, 1:35AM) to set Marty's destination time when he's returning to 1985. (01:42:40)
Continuity mistake: When Biff chases Marty out of the cafe in 1955 and Marty 'makes' the skateboard, there is an odd shot of a tree. When Biff's car drives through the grass, the big tree in the shot is shaking violently, yet all the trees are completely still during the chase before and after this shot. During this entire scene, differences regarding varying degrees of windy then calm, wet vs dry pavement and variations in shadows cast are numerous. (01:06:30)
Continuity mistake: In the 1955 skateboard chase scene around the town square, note how the skateboard used throughout the whole chase scene changes at least 4 times. Namely different wheels, trucks and the trucks' wooden mountings. (01:03:15)
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