Continuity mistake: As the Delorean accelerates towards Doc and Marty in the car park, in a shot from the car's perspective they're a few feet apart, then we cut to a close up, and they're right next to each other. (00:21:45)
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)
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: The time machine is gone for 1 minute and 21 seconds - not just 1 minute. (00:22:00 - 00:23:20)
Revealing mistake: After Einstein's time travels, when both flames cross Marty and Doc's legs, they cast no shadow, giving the feeling that they are floating, thus giving away the blue screen effect. (00:22:01)
Revealing mistake: As the DeLorean's vanity plate spins around in place, you can see how it's attached to a short steel post, stuck into a hole in the pavement. (00:22:10)
Continuity mistake: Near the beginning, right after Einstein time travels, Doc walks up to the time machine and says it's cold. He then opens the door and greets Einstein. In this split second you can see Einstein's stopwatch has a non-illuminated LED display, but in the next shot the face has changed to the bright red numbering. (00:23:10)
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 Einstein comes back Doc Brown is demonstrating to Marty how to set the time machine. He simply pushes the buttons on a key pad. To set it for July 4, 1776 he punches 3 buttons. To set it for December 25, 0000 he punches 6 buttons. To set it for November 5, 1955 he punches 12 buttons. Later, when Marty is resetting the time machine to go back 10 minutes early he punches 7 buttons (all not including the ENTER button on the keypad). (00:23:50)
Continuity mistake: As Doc reacts to the icy DeLorean, the door Doc was about to pull open barely has any ice on it, but when Doc opens the door with his foot, more ice is on it, then melts after Einstein runs back into Doc's truck. (00:24:00)
Continuity mistake: At Twin Pines Mall, when Marty is recording Doc Brown, Marty is raising the video camera to record Doc Brown saying, "No, no, no, this sucker's electrical," but when this piece of film is played back in the Doc's garage in 1955 it appears as if Marty was already recording when the Doc starts speaking. (00:25:15 - 00:50:40)
Visible crew/equipment: When Marty asks if the time machine is nuclear, a boom microphone is visible in the right windows of Doc's van. (00:26:28)
Continuity mistake: When Marty is filming the Doc in the parking lot, Doc says that he needs '1.21 gigawatts' to power the time machine. Marty then says 'Doc, you don't just walk into a store and buy plutonium.'. However, when Marty and the Doc of 1955 are listening to it on his TV, the Doc keeps talking after he says '...of electricity I need'. (00:26:35 - 00:53:05)
Visible crew/equipment: A boom mic reflection is visible on the vent window of Doc's van when Marty asks him if the plutonium was ripped off. (00:26:35)
Continuity mistake: As Doc turns the plutonium jar on the back of the DeLorean, the little thing sticking out of the jar slides greatly down, without any motion of moving. (00:27:15)
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: 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: 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)
Other mistake: When the Libyans shoot at Doc Brown and the bullets hit his van they leave no bullet holes, just impact sparks. (00:29:10)
Continuity mistake: When Marty is being chased by the Libyans, a few times it cuts away to show the instrumentation in the car. The first time the odometer reads 33064.2 and the trip counter says 88.8; the second time it's goes back a little to 33061.8 and 86.4. A few seconds later we see it as he turns the corner, and it goes down again from 33,061.8 to 32,994.4, the trip counter has gone from 86.4 to 19. In fact it's a completely different set of instrumentation - the bottom of the needle's angled in some shots and flat in others. (00:29:20)
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