Question

what is the outcome martin Luther king wants out of writing the letter of Birmingham?

Answer 1

The letter defends the strategy of nonviolent resistance to racism. It says that people have a moral responsibility to break unjust laws and to take direct action rather than waiting potentially forever for justice to come through the courts. Responding to being referred to as an "outsider," King writes, "Injustice anywhere is a threat to justice everywhere".
https://en.wikipedia.org/wiki/Letter_from_Birmingham_Jail

Answer 2

Maybe this link will help http://www.history.com/news/kings-letter-from-birmingham-jail-50-years-later :)