Martin Luther King, Jr. had more to do with the Brown case than Malcolm X did. Neither man was directly involved in the litigation, but King had more to do with trying to extend the reach of the Browndecision than Malcolm X did.
In Brown, the Court said that segregation in public schools was illegal. This enshrined the idea blacks and whites should be equal and society should be integrated. King's major work was to try to push for that to truly happen. He was trying to make sure that segregation would truly end in all areas of society. That was a large part of what the Civil Rights Movement was about. This goal was realized in 1964 with the passage of the Civil Rights Act of that year.
Malcolm X had less to do with this process. He did not believe in integration. He wanted African Americans to progress and be strong, but he did not want them to do so by integrating with whites. He was in favor of black nationalism. Therefore, he was less involved in extending the logic of Brownthan King was.