(Spoilers to follow, though I'm not sure that really applies when you're just talking about presented history, especially so long after the film's release.)
Very interesting to see so many familiar names, not just from history but from contemporary politics, names like Andrew Young and John Lewis. I don't know the history well enough to know how accurate it was, but as story it worked fairly well. I mean, pretty much every one had their motivations and flaws, and those all made sense in terms of their history. You had lots of competing factions who thought the march was a good or a bad thing - Malcolm, the SDCC, Johnson and J. Edgar and the infamous "white moderates" - and each of their positionsactually made sense for both personal and political reasons. It had a lot of the feel of old-school "Law and Order," the dialectic where the principals all disagree and they're all right or at least reasonable because reality's just that complicated.
I would have liked to see some things expanded on, like Coretta's psychology. There's King's supposed infidelity, which is raised as an issue (you have J. Edgar sending a sex tape of MLK with another woman to Coretta), and I found myself wondering what their marriage must have been like. Would infidelity have been worse than putting their children in danger through his activism? I wanted to know more about that dynamic, why she stayed, why she reacted the way she would. But I guess that's where fanfic (or historical fiction) enters in a bit.
For me, the biggest problem was the way it treated King's white allies. There are all kinds of white clergymen who joined the march. That's historically true, and I'm sure in 1965 it felt necessary, probably even a bit of a boon: here are white people getting involved in the struggle, taking an active role and not acting like the white moderates who called for patience and delay. As a white person I'm not sure how I would have acted any differently in that situation, what other path would have been open to me. But at the same time, there's a sense of white savior complex, of white people being necessary to win the day. That's probably historically accurate, but there's no real questioning of it (except the practical point, that it won't last long). In 2016 that bothered me.
All in all a very nice movie and worth watching if you haven't already.