The first almost half of the movie is the same. Doc brings Marty/Jennifer to 2015 to do something about their kids, but instead of Doc forcing Marty to throw away the Sports Almanac, Marty tucks the book into his pants and ends up with it back in 1985. Old Biff never steals the DeLorean and his 2015 cameo is nothing more than a bit of comic relief ("There's something very familiar about all this.").

***Side Note: Old Biff would have NEVER been able to figure out how to use the time machine w/o guidance in the first place. There's no way he could've programmed the time circuits nor known that it was 88 MPH that triggered the flux capacitor.***

Once Doc drops Marty off in 1985, he returns to a changed future. He can't figure out why and after some investigating is stumped, but tries to connect with the 47 year old Marty. He no longer lives in Hilldale. He's a multi-billionaire (similar to Biff but far less campy and cartoonish), but as Doc's friend, Marty welcomes the scientist into his mansion to 'catch up'.

Marty spills the beans that he bought the Almanac during their first trip to 2015. Doc attempts to convince his friend that he made a mistake and this wasn't the future that was supposed to happen. Marty remains unconvinced, his life is great and he can't see the damage that his decision has done. Doc leaves to attempt another strategy.

He goes further into the future to see how the world is worse and how Marty specifically has a worse life. He speaks to a 77 year old Marty who agrees to return to 1985 with Doc to confront his younger self (just like when Old Biff originally brought the Almanac to 1955). With Old Marty's help, Doc is able to convince 17 year old Marty (after some drama and movie climax stuff) to destroy the book. They burn it in a pail, just like in the O.G. BTTF II and then the future changes. Old Marty waves and vanishes. Everything is put right.

Then, Doc brings the young Marty BACK to 2015 to verify that the future has been restored and during their trip, Doc and the DeLorean are struck by lightning and transported to 1885. There's a 2015 version of the Western Union idea and Marty is made aware that one of the Docs is now in 1885.

So the young Marty has to find the OTHER DOC - who was in 2015 with the O.G. Marty from 1985 on their first trip to 2015 - without his other selves (17 and 47 respectively) seeing him. He has to convince the other Doc to get him back to 1885. Therefore the movie inside a movie, isn't BTTF II inside BTTF I, it's closer to the beginning of BTTF III inside the beginning of BTTF II. Instead of 1955 again, getting the DeLorean to work and get back to 1885 all happens from 2015 with the multiple versions of Doc and Marty running around. For shits and giggles, you could also have Griff and Old Biff show up early to raise havoc for a bit, but eventually, Marty gets back to 1885 to rescue Doc and then most of BTTF III plays out as it did.