How does MLK Jr. combat racism?

Dr. Martin Luther King Jr. wrote what we now know as "Letter from Birmingham City Jail" as a response to eight religious leaders (white men) who were expressing concern about the nonviolent protests Dr. King was leading. Dr. King was imprisoned in Birmingham because of his participation in the protests. In this letter, Dr. King combats racism in many ways. 
The first way he combats racism is to address it and call it what it is. Dr. King was far from being politically correct! He was bold to call the treatment of black citizens unjust, and he spoke the truth that others wanted to deny. 

I would not hesitate to say that it is unfortunate that so-called demonstrations are taking place in Birmingham at this time, but I would say in more emphatic terms that it is even more unfortunate that the white power structure of this city left the Negro community with no other alternative. IN ANY nonviolent campaign there are four basic steps: collection of the facts to determine whether injustices are alive, negotiation, self-purification, and direct action. We have gone through all of these steps in Birmingham. There can be no gainsaying of the fact that racial injustice engulfs this community. Birmingham is probably the most thoroughly segregated city in the United States. Its ugly record of police brutality is known in every section of this country. Its unjust treatment of Negroes in the courts is a notorious reality. 

Dr. King pulls no punches. He is in Birmingham because racism is there, and his purpose is to stand against injustice in all its forms. 
Another way Dr. King combated racism was by breaking laws. This is sometimes called civil disobedience. In his letter he gives his rationale for doing this: 

You express a great deal of anxiety over our willingness to break laws. This is certainly a legitimate concern. Since we so diligently urge people to obey the Supreme Court's decision of 1954 outlawing segregation in the public schools, it is rather strange and paradoxical to find us consciously breaking laws. One may well ask, "How can you advocate breaking some laws and obeying others?" The answer is found in the fact that there are two types of laws: there are just laws, and there are unjust laws. I would agree with St. Augustine that "An unjust law is no law at all." Now, what is the difference between the two? How does one determine when a law is just or unjust? A just law is a man-made code that squares with the moral law, or the law of God. An unjust law is a code that is out of harmony with the moral law. To put it in the terms of St. Thomas Aquinas, an unjust law is a human law that is not rooted in eternal and natural law. Any law that uplifts human personality is just. Any law that degrades human personality is unjust.

King gives a further illustration of just and unjust laws in this letter. He reminded his readers that everything Hitler did in Nazi Germany during the Holocaust was legal and that everything the Hungarian Freedom Fighters did to combat Hitler was illegal. 
Dr. King combats racism by calling on Christians, who are united in Christ, to come together for the cause of combating injustice. He speaks about his disappointment in the religious leaders and white community of believers because they have not stood with their black brothers and sisters in fighting injustice. 

I hope the church as a whole will meet the challenge of this decisive hour. But even if the church does not come to the aid of justice, I have no despair about the future. I have no fear about the outcome of our struggle in Birmingham, even if our motives are presently misunderstood. We will reach the goal of freedom in Birmingham and all over the nation, because the goal of America is freedom. Abused and scorned though we may be, our destiny is tied up with the destiny of America. Before the Pilgrims landed at Plymouth, we were here. Before the pen of Jefferson scratched across the pages of history the majestic word of the Declaration of Independence, we were here. For more than two centuries our foreparents labored here without wages; they made cotton king; and they built the they built the homes of their masters in the midst of brutal injustice and shameful humiliation—and yet out of a bottomless vitality our people continue to thrive and develop. If the inexpressible cruelties of slavery could not stop us, the opposition we now face will surely fail. We will win our freedom because the sacred heritage of our nation and the eternal will of God are embodied in our echoing demands.

Dr. King also talks about being an extremist, which he at first resisted but then embraced. He is an extremist because he is willing to do whatever it takes to gain equality for the people he is leading. This is another way he combats racism—his passionate commitment to his cause. His commitment was unwavering in the face of opposition, threats, setbacks, and discouragement.