Continuity mistake: When Marty is at the "Start" line waiting for the alarm clock to ring, he moves up his return time and the engine stalls out. Next we see Marty turning an ignition key with several other keys on a chain like the ones used on ceiling fans with the little balls and a plastic fob with the paper insert that reads 555-1138 Out-A-Time on it. After Marty arrives back In 1985 in town square, he pauses and gets out for a moment. While returning the the car he puts the DeLorean in drive and again the car stalls out. Again we see Marty restarting the DeLorean, we see a different set of keys. The key mysteriously changed in the time travel to an after market type with a thin wire ring on it, no key fob and the other keys are missing. (01:38:43 - 01:42:44)
Continuity mistake: When the Doc is trying to pull the cable to connect it, the cable is stuck. And when he pulls the cable hard, the other connection breaks off. Watch when it breaks off, you can see that after Doc pulled it, the cable gave him more length and would surely get Doc's arm to the other cable. When they show the Doc a second after, he hasn't moved one bit. (01:39:35)
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)
Continuity mistake: When Marty is playing at the dance the picture on the guitar disappears in some shots.
Continuity mistake: After Doc accidentally unplugs the cable and is sending Marty back, he unplugs the cable 15 seconds later and it shows Doc, and the clock's hand change. A split second later the clock's hand changes again.
Visible crew/equipment: When Marty is riding past the cafe on his skateboard and all the chicks are waving at him, you can see the reflection of the camera lorry and crew behind him. (00:06:25)
Continuity mistake: While being chased by Biff, after Marty knocks down a man and a woman, the man stands up and helps the woman stand up, but a frame later he is rolling on the floor with the woman on top of him.
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: After the shelves fall on top of Marty, the rubbish on top of him changes between shots. (00:04:10)
Audio problem: Lorraine comes to see Marty in Doc's garage in 1955 to ask him if he will go to the dance with her. Just before Lorraine moves to the other side of the car from where she walked in, she begins to say something to Marty. The problem is that her lips don't move when she talks, I guess the right clip ended up on the editing room floor. (01:11:40)
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)
Continuity mistake: Marty is hitching a ride on the back of a jeep, a wide shot shows him going past and beyond the 'Lou's Aerobic Fitness Center' next to the Texaco garage, there's a cut to his skateboard then back to a medium shot of him going pass the same fitness centre again, this time waving to the people inside. (00:06:00)
Revealing mistake: When Doc starts up the DeLorean watch the remote when he says to Marty "not me, the car, the car." The counter is static at 00 but the car is moving. (00:19:40)
Continuity mistake: Cafe 1955, when Marty enters for the first time. The arrangement of the coats on the rack to the right-hand side of the cafe doors changes upon Marty's initial entrance and returning to the counter after using the phone booth. (00:35:50 - 00:36:40)
Continuity mistake: In the cafe, the menu in front of Marty changes after Biff leaves. It starts looking like an ice-cream menu, with swirly lines at the top, but just before the future mayor comes into shot, the design on the front changes completely. (00:38:15)
Continuity mistake: In the scene where Marty tells George his plan for getting Lorraine to go to the dance with him, Marty's shirt is sometimes tucked in and sometimes not. (01:12:40)
Other mistake: The first time-travel experiment involving Einstein the dog, he goes one minute into the future after the DeLorean excelled to 88 mph. Yet after the minute is up and Einstein returns, the car emerges already braking, and screeches straight to a halt. At 88 mph, it would have to travel much further even to quickly stop the car. Look at all other time-changing moments in the three films. The car takes much longer to stop.
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)
Character mistake: At the beginning of the movie Marty is turning on a huge guitar amplifier. The two switches for the filaments are misspelled. They are shown as "filiment." (00:03:25)
Continuity mistake: As the Peabodys try to determine what crashed into their barn, the DeLorean's left indicator flashes throughout. When the family is onscreen, the barn's interior goes completely dark between flashes - whereas there should be illumination from the headlights as well. Compared to a later point in time where Marty opens the barn door to get shot at, it's not hard to notice how bright the headlights are. (00:31:50)
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