How did MLK Jr. see a relation between progressivism and the Civil Rights Movement?

King's ultimate goal was the liberation and full citizenship of all Americans. However, he knew that other pursuits -- class and gender equality -- were not unrelated to efforts to desegregate the nation.
King is best remembered for his efforts to desegregate buses (the Montgomery Bus Boycott of 1955), to end obstructions to black people voting in the South (the march to Selma), and fair and equal access to housing (his march in Cicero, Illinois). 
The March on Washington included people of all racial and class backgrounds. Its organizers were also diverse. Very often, when we discuss the Civil Rights Movement, we focus mainly on the work of black men who were not homosexual. However, the movement included women -- not only Rosa Parks, but also Claudette Colvin and Pauli Murray. It also included gay men, such as James Baldwin and organizer Bayard Rustin. King's willingness to work with everyone is an indication that he saw the civil rights effort as intersectional -- that is, cutting across racial, gender, sexual orientation, and class lines.
When he was assassinated in Memphis on April 4, 1968, he was in the city to help organize sanitation workers. Given the location of this effort, King was not only seeking to improve conditions for all of these low-paid workers but was probably also addressing the concerns of low-income workers of color, given that they were most likely to be employed as sanitation workers.