Continuity mistake: After George knocks Biff behind Marty there is a couple running to George, but in the open view they disappear.
Continuity mistake: After Marty gets to the square in 1955 his hair changes after he grabs the newspaper.
Continuity mistake: When Marty arrives at 1955-Lyon Estates he removes his hood and his fringe is brushed from right to left. When the angle changes it has swapped sides.
Continuity mistake: Right before Doc presses the remote control to send Einstein to the future, his left index and thumb fingers are on the remote. In the close-up it's just his thumb on it. In further close-ups it swaps to the index and thumb.
Continuity mistake: Before Doc sends Einstein to the future and says "If my calculations are correct...", Marty raises his left arm. When the angle changes it's lowered.
Continuity mistake: While Marty is talking with Jennifer in the square he is holding a tape case in his left hand which disappears in the next shots.
Continuity mistake: While Doc is explaining to Marty how the machine works, he has a device on his index finger on his right hand that moves in the next shots.
Continuity mistake: When Marty drives out of the Peabody barn after arriving in 1955, it's night, or certainly pre-dawn - the sky is very dark from every angle when he's fired at while driving away. Next shot he drives past the "Peabody ranch" sign onto the main road and it's significantly brighter, with the sun rising over the horizon. Then when he notices the Lyon Estates sign shortly afterwards it's broad daylight - judging from the length of the shadows the sun's very high in the sky. Lyon Estates isn't that far from Twin/Lone Pine(s) mall.
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.
Continuity mistake: When Marty is on a skateboard being chased around the town square by Biff and his friends, he grabs the back end of a truck. He does with two hands at the same point, then uses one hand in the next shot, then two hands again.
Continuity mistake: When George walks to Lorraine with the flirting words written on a notepad, her purse changes positions between one shot and the other.
Character mistake: In the parking lot when Marty begins to film, Doc says the date and exact time according to his watch. Marty then looks at his watch. Apparently his is off because he shakes his wrist and puts the watch to his ear to see if it is still working. However, he is wearing a digital watch which wouldn't tick or likely have broken parts to rattle.
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