Character mistake: At the beginning of the movie Marty is turning on a huge guitar amplifier. The two switches for the filaments are misspelled. They are shown as "filiment." (00:03:25)
Character mistake: Before playing Johnny B Goode Marty says it's a blues riff in B, but then plays the song in B-flat.
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