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)
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)
Audio problem: Marty's singing during the "Johnny B. Goode" number is obviously dubbed; it goes out of sync several times. (01:27:40)
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)
Continuity mistake: In 1955, when Doc Brown pulls Marty's letter out of his coat pocket, he either holds the envelope at its short side edge or at its long top edge, depending on the angle of the shots. (01:32:20)
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: After the clock tower's ledge breaks and Doc's hanging off the edge, we see a shot from below, then of Doc looking straight up at the clock, which changes to 10:03. But look at the clock in the below shot: the minute hand ticks over behind him, then moves again in the close-up. (01:34:00)
Continuity mistake: At the clock tower, after the tree limb falls, Doc Brown grabs the loose, messy bundle of rope and instructs Marty, but in the next shot when Doc runs up the stairs, the rope is tied up in a rather tidy bundle. This rope bundle changes yet again as he rushes up and then reaches the top. (01:34:10)
Visible crew/equipment: When Doc Brown loses his footing on the clock tower as it begins to toll, in the next shot, when he swings about towards the clock the thin black stunt cable attached to his waist, that leads around the neck of the statue, is visible. Do not confuse it with the thick silver electric cable Doc is holding. (01:35:25)
Continuity mistake: When the clock tower bell sounds a third time, Doc backs away from the ledge and stares down below in shock. Doc's arms are hanging down at his sides, but once the shot changes he's seen cupping his ears. (01:35:35)
Continuity mistake: When the clock tower tolls at 10:00, Marty rushes to the DeLorean and as he drives off there are two huge cable spindles at the curb's red edge, near the cloth (lying on the street) that covered the DeLorean. In the next overhead shot, with Doc Brown on the ledge, the cable spindles are gone down below. (01:35:50)
Continuity mistake: In 1955, when Marty drives the DeLorean to the 'start here' line that Doc painted on the street, the line and large white lettering are faded and filthy, but when Marty gets out of the car to insert the hook pole, the line and lettering are bright and immaculate. When he finally starts the DeLorean they're back to faded and filthy. (01:36:30 - 01:38:55)
Continuity mistake: In 1955, when Marty drives the DeLorean to the large 'start here' line that Doc painted on the street, in the first shot its front is up to the line completely covering the word 'here', with some distance to the last 't' in 'start'. Its position changes in following shots, such as when Marty tries to start the stalled car in the exterior shot, the DeLorean's front end is further back with the word 'here' perfectly visible. (01:36:30)
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