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.
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.
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)
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)
Continuity mistake: When Biff and gang are chasing Marty on the skateboard, they eventually crash into the rear of a manure-hauling dump truck. When the car swerves left in an attempt to miss it, it will obviously pull somewhat to the left. In the shot just before impact, you can see that just the rear half of the car should impact the truck, but in the next shot the car rests perfectly centered behind the dumps open load. (01:07:25)
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)
Continuity mistake: When Marty gets hit by the car and wakes up in Lorraine's bedroom. She constantly calls him Calvin. Marty says "Calvin, why do you keep calling me Calvin"? When he says this he puts his arm on his lap. In the next shot behind him when Lorraine replies, his hand/arm is by his side. (00:43:40)
Continuity mistake: When George is in the school cafeteria talking to Marty about science fiction stories, he holds the notepad differently with each changing shot. (00:58:55)
Continuity mistake: While Marty's 1950s family is having dinner, the milk bottle's level is one mark below the red printed label. A second later it's 5 centimeters above the red mark.
Continuity mistake: When Doc puts Einstein in the car, the dog turns his head inside the car. A frame later, from a new angle, it's outside the car.
Continuity mistake: When Marty was recording, Doc never hesitated between "reaction" and "to", then when they are watching the tape, he hesitates a little.
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