First a caveat that I haven’t read all the books I’m recommending, and I’m going off books I’ve been interested in reading or have been recommended to me too.

For MLK specifically, there’s America in the King Years, a 3-book series by Taylor Branch that covers 1954-1968. There’s also the Autobiography of Martin Luther King Jr., compiled from his writings by Clayborne Carson.

For readings on other black leaders of the 1950s/1960s, there’s the Autobiography of Malcolm X by Malcolm X and Alex Haley; Revolutionary Suicide by Huey Newton; another Autobiography, this time by Angela Davis; and The Fire Next Time by James Baldwin. Baldwin wrote frequently on growing up and living as a black man in America so if you’re interested I’d say explore his other works too.

For works in general on the Civil Rights Movement, there’s The Race Beat by Gene Roberts and Hank Klibanoff about the media and the movement and Freedom Summer by Bruce Watson about the “Freedom Summer” of 1964.

Finally while it’s not actually on MLK or civil rights, I believe an essential reading on race, policing, and incarceration since the 1960s is The New Jim Crow by Michelle Alexander.

Hope this helps!

I'd also recommend Ralph David Abernathy's And The Walls Came Tumbling Down.

A surprising amount of people under 35 who are otherwise well-educated in the civil rights movement have never even heard of him, but Abernathy's role as the defacto second-in-command behind MLK meant that he was present at almost every major event during it. Since he lived, his autobiography provides one of the few birds eye views of not just how the movement succeeded in the 50s and 60s but how it fell apart in the 1970s.

One probable reason for his relative anonymity nowadays is that upon publication, he and the book were condemned by almost all surviving members of the movement for being the first close associate of MLK to confirm that he had multiple affairs; the one story that really got under people's skin was his allegations of what MLK supposedly was up to the night before his assassination, which led to bitter accusations against Abernathy where he was accused of selling out for the proverbial 30 pieces of silver and being mentally incapacitated due to a couple of strokes.

It's a shame, as the book was so much more than that; his despair while at the Poor People's March on Washington when he finally realized that poverty just wasn't going to be enough of a uniting factor to overcome racial differences and continue the movement is one of the more poignant moments you'll ever find in an autobiography.

And that rather than all of MLK's hijinks is why I recommend it.

Edit: Also, one media reference. If you have access to HBO, their 2018 documentary King in the Wilderness is terrific. It covers the last 18 months of his life, which is an area that most books on the movement don't spend much time on: King's break with LBJ, his growing frustration as he butted heads with racism in the North that was far more difficult to confront, and that at the time of his death the polls were showing his approval rating in the mid-20% range.