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 is looking in awe at the 1950's records store, his hair swaps from messy to brushed backwards right when he walks backwards on the grass.
Continuity mistake: After Doc Brown sends Einstein one minute into the future, the rest of the scene until he returns is played out in real time. There are no scene cuts, and any camera angle changes are accompanies by continuous audio which negates any possibility that time has been stretched or compressed. However, it takes Einstein 1 minute 21 seconds to return.
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: Whilst George McFly is pegging out the laundry, Marty is explaining his plan as to how he can make Lorraine fall in love with him. At the start of the scene, both of Marty's breast pocket flaps are untucked. After putting the bag down, one tucks itself in, then they return to normal when Marty says "let's go over the plan again". (01:09:55)
Audio problem: When Marty is playing the intro to "Johnny B. Goode" you can hear some cymbals in the background. But you can see behind him that the drummer isn't hitting any. (01:27:30)
Other mistake: When young George and Marty are in the backyard talking about what will happen at the dance, young George hangs up clothes to let them dry, but if you look closely the clothes are actually already dry.
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)
Continuity mistake: While Marty is admiring the new truck, Jennifer walks onscreen. Once the angle changes, Marty's left arm is resting on an entirely different part of the door. (01:49:50)
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)
Continuity mistake: When Marty skates off after knocking over the couple, the car behind him is red and cream, but in the next shot, it turns completely red. A yellow car also appears in the second shot, when it was not present in the first. (01:07:10)
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)
Character mistake: Before playing Johnny B Goode Marty says it's a blues riff in B, but then plays the song in B-flat.
Continuity mistake: In the whole where When Biff is chasing Marty (on the 'skateboard') around the Town Square, the sun changes at every shot.
Continuity mistake: When Doc finds Marty's letter in his pocket, the big hook fastened to the DeLorean's passenger side is seen facing the sky. Pretty soon Marty slides across the hood and into the DeLorean, but now at this point it's facing the ground instead. (01:33:35 - 01:35:50)
Continuity mistake: When the McFlys are together in their kitchen, Lorraine asks everyone to drop Uncle Joey a line. At this point George lowers his left hand and seemingly reaches for silverware on the table. But between shots, he lifts his hand back to its original position all of a sudden. (00:15:00)
Continuity mistake: Before Lorraine brings Joey's cake, George leaves a bunch of crumbs on the table and starts to watch TV and do some numbers. After the cake lands on the table, the place is spotless.
Continuity mistake: When Doc syncs his stopwatch and the dog's, he holds them away from their bodies. A frame later his hands are close to his chest and there's dog hair all around.
Continuity mistake: After Joey's cake lands on the table, George puts his arm to the right of the cereal package. A frame later it's on the left side and holding a pencil.
Continuity mistake: After escaping from the Peabodys, Marty stops the car in the middle of the road and exits with his hair all messed up. A frame later it's brushed. (00:33:57)
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