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: When Marty chases after Biff on the borrowed 'skateboard', Marty is wearing a dark grey belt and a red/blue print shirt under his red/beige jacket. However, when Marty is hanging on to the front of Biff's car as they turn a corner (and in another shot), Marty (stunt double) is wearing a light brown belt and solid tan shirt. (01:07:00)
Continuity mistake: When Biff and his goon friends are in Biff's car, as they chase Marty on his borrowed 'skateboard', the car's rearview mirror repeatedly disappears and reappears, and the side mirror changes from round to square repeatedly. (01:06:50)
Other mistake: The car that almost runs Marty over has the vehicle registration plate of 6S 48405. Later on, when Marty is hit by Sam, Sam's car has the same plate. (00:35:49 - 00:42:02)
Visible crew/equipment: When Doc Brown loses his footing on the clock tower as it begins to toll, in the next shot, when he swings about towards the clock the thin black stunt cable attached to his waist, that leads around the neck of the statue, is visible. Do not confuse it with the thick silver electric cable Doc is holding. (01:35:25)
Continuity mistake: When Marty is trying to escape from the van he turns the time circuits on. However, the time circuits were already on - you can see them illuminated when he first puts the car in gear. (00:28:45)
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: 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)
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)
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 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: 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: 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: 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)
Continuity mistake: George McFly's holding a pen in his right hand, then a cut to a wide shot reveals him not holding a pen as he rests his arm away from his body on the table. Then when the camera cuts back his arm is back in front of him holding the pen. (00:14:10)
Revealing mistake: When the DeLorean is about to time travel for the first time, the fuel trails to achieve the following fire trails are already visible on the pavement. (00:22:00)
Continuity mistake: During the Johnny B. Goode song, the mike cable's position keeps changing depending on the shot, despite no-one touching it. (01:27:30)
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