Continuity mistake: When we first see the machine in action, the travel is shown from three different angles. In the second, a line of fire passes directly between Doc's legs, but in the third angle, they pass to the left of both of his legs. (00:21:00)
Continuity mistake: In 1955, when Marty drives the DeLorean to the 'start here' line that Doc painted on the street, the line and large white lettering are faded and filthy, but when Marty gets out of the car to insert the hook pole, the line and lettering are bright and immaculate. When he finally starts the DeLorean they're back to faded and filthy. (01:36:30 - 01:38:55)
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)
Trivia: Eric Stoltz was originally cast as Marty. After filming quite a few scenes they realised his acting style was too dramatic for the humor desired, so they cast Michael J Fox (who they couldn't originally get because he was busy with the TV show Family Ties). Filming was on weekends and nights around his TV schedule and using his double at other times.
Continuity mistake: When Marty is writing the letter to Doc before he goes into the future, the word "disaster" is the only word in the last line. Yet when Doc shows Marty the letter when it has been all taped up, "terrible disaster" is the last line. Also, the handwritring in each letter is different (i.e. Marty's signature). They are two different letters. (01:40:55)
Continuity mistake: When Marty is standing over his father's bed in 1955 persuading him to date Lorraine, there is a hair-dryer in Marty's belt that seems to disappear between different frames (note the hair dryer's cord leading to Marty's right hand, as he holds it behind his back). Really obvious. [This scene was originally longer, including a part where Marty removed the hair dryer from his belt. Still a mistake, but there's the explanation.] (00:59:45)
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)
Continuity mistake: When Marty is in 1955, he is in shock and walking on the pavement when a car comes by. In the first shot when it comes towards him it has two speakers, on the second shot when it's moving away, it only has one. In the second shot of the car with only one speaker, the A-frame sign is also turned the other way. (00:35:15)
Trivia: In the first scene when Marty arrives at Doc's place and the camera pans across all the clocks, one particular clock has an hour hand and a minute hand with a man hanging from it - foreshadowing Doc's perilous scene hanging from the clock tower. The hanging man isn't Doc, but the silent movie star Harold Lloyd, in a reference to him in Safety Last. Sadly though, Harold and Christopher aren't related. (00:01:35)
Continuity mistake: When Marty's coming home at the beginning of the film, he's watching the discussion between his father and Biff who complains about the wrecked car and his reports. In the two shots which show Marty, there's a glass full of candies near to him. When Biff steps to the glass and lifts the lid, half of the candies are gone. (00:12:15)
Continuity mistake: When Marty is standing over his father's bed in 1955, Marty puts the headphones on George and then puts the tape in the cassette player. If you notice the hairdryer is on the left side of the belt and is touching his left arm, but when they cut to Marty holding the tape player out in front of him the hairdryer is on the right side. (01:02:10)
Continuity mistake: In one shot of Einstein strapped in the DeLorean, if you look closely in between the steering wheel, the yellow time circuits on the bottom read the month as "10" rather than "Oct." And the red and green time circuits read "11" instead of "Nov." (00:19:20)
Trivia: The amplifier that Michael J Fox blows up at the beginning is labeled as being a CRM-114. This is a reference to Dr. Strangelove, in which the B-52 crews receive their orders over a CRM-114. It was also the serial number of the Jupiter explorer in 2001: A Space Odyssey (1968) also directed by Stanley Kubrick. (00:03:05)
Continuity mistake: At the beginning of the movie after Biff leaves the McFly home. There is a scene where Marty and George are talking. In one shot George has three pens in his pocket, a few seconds later he has two. (00:14:14)
Continuity mistake: When George goes into the cafe to tell Lorraine he is her density/destiny, the people sitting at the bar keep changing with different shots. Best example is just after George refers to his "density" - in the background there's a brunette in a blue jumper at the bar. A blonde woman and a short-haired guy then sit on her left, but then disappear in the shot of Marty sitting at the bar - on the brunette's left there's now another brunette in a pink jumper with a different guy. The brunette's got shorter hair too. In the very next shot the blonde's back with no sign of Marty. (01:02:10)
Continuity mistake: At the Twin Pines mall during the chase scene with the Libyans, there is one shot where you get a great look at the driver as he's trying to start the engine. Later on you get another look at the driver as the van is moving and it's a totally different person. (00:29:52)
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)
Continuity mistake: 1955 Skateboard Chase Scene around Town square. After Marty falls over the couple, a shot of Biff's car shows all the passengers standing up, but in the very next frame everyone is sat down. Also, that is not Biff driving. (01:04:20)
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