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 car slams into the manure truck, in the first shot the side of the truck is adjacent to the car's back passenger door, but in the next overhead shot the side of the truck is adjacent to the front windshield. (01:07:20)
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: After Goldie Wilson arrives at the manure truck scene, we can see onlookers approaching at the street corner in front of Texaco - one of which is the kid who Marty borrowed a skateboard from. But once the shot changes, that same kid is instantly standing on the sidewalk near Marty holding a crate instead. (01:07:35)
Continuity mistake: When Biff and his friends are chasing Marty on the skateboard, just before they hit the manure truck there is canvas covering the fertilizer which disappears in the next shot. (01:07:45)
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: In the scene where Marty tells George his plan for getting Lorraine to go to the dance with him, Marty's shirt is sometimes tucked in and sometimes not. (01:12:40)
Continuity mistake: When Marty takes a swig from the bottle, we see the motion of Lorraine's hands as she lights a cigarette. She finishes lighting and then grabs it with her left hand. In the next shot where Marty spits alcohol all over the place, Lorraine is busy lighting her cigarette a second time. (01:18:10)
Continuity mistake: When George is checking the time at the prom there is a girl in grey behind him that changes to one in pink. (01:18:56)
Continuity mistake: At the dance, when Marty is parked with Lorraine, Biff reaches into the car and grabs the front of Marty's shirt/tie, but in the next exterior shot, Biff now has got hold of the back of Marty's suit. (01:20:10)
Continuity mistake: After George lays Biff out, Marty pulls the family photo from his pocket. The way he holds it suddenly changes, judging by the position of his index finger as the shots change. (01:23:45)
Continuity mistake: When Marty is at the Enchantment Under the Sea dance and is about to disappear, he puts up his right hand and it starts to disappear. On the very next shot of him, he is putting down his left hand. (01:25:35)
Continuity mistake: Before his parents kiss, Marty's right hand begins to disappear. As he slumps to the floor both hands are visible holding his guitar. (01:25:35)
Continuity mistake: Enchantment under the sea dance: In addition to the comment about Strickland's hands covering his ears twice note some of the people around him have either moved or disappeared. (01:25:55)
Continuity mistake: When Martly finishes his rambunctious guitar solo at the dance the camera pans the crowd. In the distance on the left we see Mr Strickland uncovering his ears. After this they show a close up of him uncovering his ears again. (01:25:55)
Continuity mistake: When Michael J Fox's parent kiss and the family picture (with everyone standing on the grass) gets fixed, Michael places the photo on the guitar, gets very happy and starts playing the guitar enthusiastically. Seconds later, a brief shot of the guitar shows the picture with Marty only, his brothers are missing. (01:26:18)
Continuity mistake: During the Johnny B. Goode song, the mike cable's position keeps changing depending on the shot, despite no-one touching it. (01:27:30)
Continuity mistake: When Marty puts the letter in Doc Brown's jacket pocket he puts it in the right one (from the way the pocket is slanted, plus we see the buttons right next to it) but when Doc pulls it out it is in the left one. (01:29:40)
Continuity mistake: While Doc is waiting for Marty to get back from the dance, he looks at the watch on his left hand, his pocket watch, and the watch on his right hand. When he looks at the watch on his right hand, he is wearing it on the inside of his wrist. Then, as he is running around the car, he has no watch on his right wrist at all. Moments later, as he is pointing down the street, it is on the outside of his right wrist. When they are checking the alarm clock, he holds it in front of the alarm clock to check the time, and the watch is still on the outside of his right wrist. (01:31:15 - 01:32:30)
Continuity mistake: After the dance, when Marty arrives at the clock tower in Doc's car (across the street from the DeLorean), he parks precisely midway between the two stores. As the clock tolls, Marty jumps into the DeLorean, and in the overhead shot as Marty drives off, Doc's car is directly in front of the store with the blue awning. (01:31:20 - 01:36:00)
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