Just looking at evidence from the movies, one really basic piece of evidence is the scene near the end of part I where Doc drops Marty off back at home, right after the scene at the mall where he showed Marty the bulletproof vest. From the transcript here, Doc says in the dialogue he's planning on making a trip "about 30 years" into the future:
Marty: About how far ahead are you going?
Doc: About 30 years, it's a nice round number.
Marty: Look me up when you get there, guess I'll be about 47.
Doc: I will.
Marty: Take care.
Doc: You too.
Marty: Alright, good-bye Einy. Oh, watch that re-entry, it's a little bumpy.
Doc: You bet.
And if you re-watch this scene you can see that after this dialogue, the DeLorean drives up the road in one direction and then Marty watches it speed by in the opposite direction, and right after it disappears from view there's a flash of bright light accompanied by the usual sounds of a time jump. So, it seems Doc immediately made his trip 30 years into the future after this dialogue, taking Einstein with him (which connects to the dialogue from part II quoted by @Acccumulation where Doc said he had left Einstein in a suspended animation clinic). And of course the next scene with Doc in part I is him coming to get Marty and Jennifer wearing futuristic clothes, so the implication is that he's just returned from the same trip to the future we saw him setting out on a few scenes earlier.
Then in part II, note that in 2015 when Doc saw that Jennifer was being taken by the police to her older self's home in Hilldale, he was extremely worried about the danger of time paradoxes, as shown in this dialogue (from the transcript here):
Doc: Great Scott! Jennifer could conceivably encounter her future self, the consequences of that could be disastrous.
Marty: Doc, what do you mean?
Doc: I foresee two possibilities. One - coming face-to-face with herself thirty years older could put her into shock and she could simply pass out. Or two - the encounter could create a time paradox, the results of which could cause a chain reaction that would unravel the very fabric of the space-time continuum and destroy the entire universe. Granted, that's the worse case scenario. The destruction might, in fact, be very localized, limited to merely our own galaxy.
After they got Jennifer back in 2015 and were about to return to 1985, Doc said "Lets get her back to 1985. And then I'm going to destroy the time machine!" When Marty asked why, he said "The risk is just too great, as this incident proves" (referring to Jennifer meeting her future self). After he dropped Marty off in 1985 (actually 1985-A, but they didn't know that), he said "If you need me, I'll be in my lab, dismantling this thing."
These scenes from part II give strong evidence that he was the Doc from 1985. If he was actually from 2015 (or any other year past 1985), and was so worried about the risk of time paradoxes, he wouldn't have wanted to dismantle the time machine in 1985, since that would leave him as a person from the future living out the next bunch of years in his own past (potentially changing things from the way he remembered 1985-2015 happening the first time around).