Continuity mistake: Marty's hairstyle changes when Jennifer kisses him on the town square bench. (00:10:45)
Continuity mistake: Before Doc's time travel demonstration, he shows Marty two synchronized clocks. Here we see Einstein turn his head towards the front of the DeLorean, but in the next frame he's facing the camera instead. (00:20:05)
Visible crew/equipment: In the opening scene, when the TV turns on as the news comes up, you can see a man with a white shirt in the reflection of the TV. (00:01:50)
Factual error: In the skateboard chase scene, after the crash, a crowd gathers. In this crowd there is a kid dressed in a Cub Scout uniform. He has a red/white town strip patch on his arm. In 1955 the Cub Scout town strip patches were blue with gold lettering. (01:07:48)
Continuity mistake: When Biff's gang arrives at the diner for the first time, Skinhead walks behind Marty and plays with a ball. When the shot changes, he is walking behind him again, but this time he is not toying with the ball. (00:38:50)
Continuity mistake: When Marty tells George how to flirt with Lorraine, they both enter the diner and Marty sits on the back of the counter but disappears between shots, only to reappear later on. Also, the extras around him change, although they're all dressed in similarly colored clothes. Check a man in a plaid jacket next to the blonde girl in pink, who disappears when Marty trips Biff. This continuity havoc was due to the combination of Eric Stoltz scenes and new Michael J. Fox reshoots.
Continuity mistake: When young George walks to Lorraine's table to read the poetic lines, the pen in front of her books moves suddenly to being on the right side of the books. (01:04:42)
Continuity mistake: When Marty enters Lou's Cafe for the first time, check a group of people gathered outside next to the doors who say goodbye and leave the place. A shot later, they're all back together, talking. (00:37:46)
Continuity mistake: At the very end of "Earth Angel", after kissing Lorraine, George waves to Marty on stage, who waves back and flexes his right hand, holding it up. It then switches to a wide shot for the end of the song, and Marty's right hand is down on the guitar again.
Continuity mistake: When Marty is holding on to the back of the truck, Biff attempts to hit him with his car. Marty quickly pulls himself to the side of the truck to avoid being hit but in the next shot, he is holding on to the back of the truck.
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)
Revealing mistake: Right before Einstein is sent to the future, when a very bright light surrounds the DeLorean, not only is the dog gone, but a black bulky bag has appeared on the passenger's seat. Seemingly a hidden stunt driver or a driving mechanism. (00:21:56)
Continuity mistake: After the DeLorean disappears with the dog inside, flames appear underneath Marty and Doc. In the close-up of their legs, a shot later, the road is spotless and the flames reappear a second later.
Continuity mistake: When Marty is up on stage "disappearing" and yells "George..." (wanting for George to resume the dance with Lorraine and ultimately kiss her), this exact shot of Marty shows him slumping over and appearing very weak, almost on his knees. A split second later showing that same scene (right when George finally walks back to Lorraine) only from the "crowd view", Marty is still standing upright, not slumped, maintaining his composure.
Continuity mistake: In the scene at the beginning of the film where Marty cranks up his giant amplifier, the resulting explosion blows Marty into a shelf which tips over, spilling all of its contents on top of him. In the next shot, despite the shelf having been emptied in the previous shot, papers and file folders continue to fall on Marty.
Continuity mistake: When Marty is playing back the Twin Pines Mall demonstration video for Doc, the camcorder tilts itself between shots, judging by the lens which sits higher above the TV set when Doc starts shouting, "1.21 gigawatts." (00:53:00)
Continuity mistake: After Goldie Wilson arrives at the manure truck scene, we can see onlookers approaching at the street corner in front of Texaco - one of which is the kid who Marty borrowed a skateboard from. But once the shot changes, that same kid is instantly standing on the sidewalk near Marty holding a crate instead. (01:07:35)
Continuity mistake: When the 1950s clock chimes and Marty stares at it, a huge scaffolding structure is reflected on a glass billboard, which is nowhere to be seen in subsequent shots of the place.
Continuity mistake: After the DeLorean with the dog in zooms in between Doc and Marty's legs, all of the angles show that there is no license plate anywhere around. Suddenly it appears spinning in front of them, as if it had been there all the time.
Continuity mistake: After Marty exits the cafe to make himself a skateboard out of two wooden crates, the crates are half a meter apart from each other, but close in the next angle. Also, the chubby kid holding his friend's crate swaps to holding his between one frame and the other.
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