What’s the claim of this letter?

The general claim that King makes in the letter is that the civil rights movement is justified in using unlawful methods to achieve its goals. In writing the letter, King is attacking those Southern white clergymen who argued that the civil rights movement should use exclusively legal means to achieve its goals. He argues that such an approach is ineffective, not least because the legal system is one of the key institutional supports of racial segregation. Under the circumstances, the only way that the civil rights movement can achieve political change is through direct action, such as staging sit-ins at segregated lunch counters and holding street demonstrations.
King also goes on to remind the Southern clergymen of their duty as Christians to challenge unjust laws. Christians cannot look the other way while their fellow men are being persecuted and oppressed. They must join with them in their struggle for freedom, even if it involves breaking the law. Only this way will it be possible to achieve the kind of just society that all true Christians should want to see established.