Revealing mistake: 1955 skateboard chase scene around town square. Note in this scene a metal device has been added between the kicktail and the back truck of the skateboard to create the sparks, then disappears and reappears throughout the whole chase scene. (01:03:40)
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)
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)
Revealing mistake: When Marty is being chased by the terrorists, when he makes the 2nd turn, if you look closely at Marty it is obvious that it is not Michael J. Fox behind the DeLorean windshield, rather it's a stunt driver resembling the original Marty, Eric Stoltz, who was replaced early in filming. (00:30:45)
Revealing mistake: When Marty is being chased on the skateboard in 1955 he skids rounds a corner and you can see that this scene had been shot before as there are skid marks there from previous takes. (01:03:20)
Revealing mistake: When Marty crosses the park in front of the court house and the camera pans around following Marty while showing the town, watch one of the buildings in the middle background. Part of the building is grey and has 2 windows and is supposed to have a slanted roof, but if you watch closely, it appears that building is in fact only a wall with nothing behind it. (00:34:40)
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)
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)
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)
Revealing mistake: When Biff and his gang are chasing after Marty in their car around the Courthouse Square, in the shots that takes place in the Courthouse yard, you can see that it's Biff's stunt double that's driving. (01:06:45)
Revealing mistake: Marty finds Doc's name in the phone book; when he sees the name, then runs his fingernail over to the phone number. Look closely at the phone book; it has several creases across the same place where Michael J. Fox ran his fingernail from previous takes. (00:36:15)
Revealing mistake: When Marty is skateboarding and being chased by Biff's car, Marty's stunt double is noticeable because his wig is bulkier and darker coloured than Marty's real hair. He looks just like the Trololo guy from Youtube. (01:40:50 - 01:41:25)
Revealing mistake: After Doc sends Einstein back in time, it shows fire trails on the pavement. Marty's leg is right in a (superimposed) flame, with no ill effects at all. (00:21:05)
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: When Doc is using a remote control to drive the DeLorean, they show a close-up of the controller. The controller's power meter (the dial with a needle in the middle) shows no power - it's right over to the left. How's he driving with no battery? (00:20:15)
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 Marty begins to slump down due to fading from existence look at the guy playing the piano behind Marty, he's not even pressing the keys on the piano just putting his palms on them repeatedly.
Revealing mistake: As the DeLorean's tyres start igniting as it is about to travel back in time, the flames are not coming from the tyres, but from some ignitors attached to the car. (01:40:36)
Revealing mistake: When the DeLorean travels back in time after Marty returns to 1985, the light emitting from the DeLorean does not actually come from the DeLorean, but from the side of the screen instead. (01:44:16)
Revealing mistake: While Biff and the others are chasing Marty on the skateboard, when Marty grabs a green truck the stunt man is doing it for him.
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