While their story was never revealed, it seems pretty clear that Marty met Doc at Hill Valley High School.
When Principal Strickland catches Marty and Jennifer in the hall and gives them tardy slips, he says “Am I to understand you’re still hanging around with Dr. Emmett Brown, McFly?” “…Now let me give you a nickel’s worth of free advice, young man. This so-called Dr. Brown is dangerous. He’s a real nutcase. You hang around with him, you’re gonna end up in big trouble.”
Being that Strickland knows who Doc Brown is, it makes many people think that Doc worked at the School, but was fired for being ‘dangerous’ or a ‘nutcase’.
There are no other hints at all in the trilogy as to how they met, but this seems to be a popular answer.
Original Quora question – How did Marty Mcfly meet Doc?