Martin Luther King Jr.'s attempts to gain equality met with slow, qualified success over time. Coming on the scene in earnest in 1955 during the Montgomery Bus Boycott, you can say that his effort to desegregate the busing system there were entirely successful, but it took nearly a year and another Supreme Court ruling in his favor for it to happen. Both that decision and the Brown vs, Board of Education decision a year earlier were not a result of any of MLK's efforts, and had a huge impact on his degree of success.
His March on Washington in 1963 was important, and historically significant, but did little to persuade our government to act any differently or more quickly for civil rights.
His "Letter From a Birmingham Jail" is considered one of his more important pieces of writing, but at the time it had little impact at all.
So I would say MLK's successes were on an individual basis, place by place, case by case, and had a limited effect on the big picture of equality at the time. We celebrate and recognize them more today because of their significance historically, while at the time, the FBI is still investigating for being a supposed communist. It was only in combination with others, and the government in specific cases, that civil rights was successful.
A major success of Martin Luther King, Jr. was the March on Washington for Jobs and Freedom. This march was on April 28, 1963 and is estimated to have attracted up to 400,000 supporters. It was at the conclusion of this march that King made his famous "I Have a Dream" speech. This march brought civil rights awareness to the forefront. It educated many Americans on the importance of civil rights. King's efforts helped in passing the 1964 Civil Rights Act and the 1965 Voting Rights Act.
A failure of King was his opposition to the Vietnam War. Although many opposed this war he placed a racial spin on it and this did not sit well with a lot of his supporters. A group of his supporters went as far as to kick out all white members and they advocated violence.
In my opinion, all of King's attempts to gain basic rights were successful. But then, when he went beyond basic rights, he started to lose.
For example, King and the movement were able to gain the right to vote. They were also able to get segregation eliminated more or less completely (at least by law). These are what I call basic rights.
Afterwards, though, King started to push for more complete equality. He tried to desegregate housing in Chicago (where the segregation was not based on law) and he tried to get more jobs in city government to be given to blacks. He failed. When he was killed, he was in Memphis, TN trying to help some garbage collectors with a strike. When he tried to push for this sort of thing, he generally failed.
When I teach about this, I always ask my students to think about why whites would have accepted King's first requests or demands and rejected his later ones. What do you think?