What various things does Martin Luther King Jr. invoke to enhance the ethos of his “I Have a Dream Speech?"

Like many black leaders and revolutionaries before him, Dr. King mainly invoked Christian ethics and the revolutionary rhetoric embraced by patriots to show that the cause of Civil Rights is a cause that is quintessentially American. To ignore the plight of black Americans was, in King's view, a failure to live up to the nation's promises and potential.
He begins by invoking Lincoln's signature of the Emancipation Proclamation in 1863 -- one hundred years before the March on Washington, at which King delivered his speech. However, contrary to Lincoln's promise, black people remained in a state of bondage. The "manacles" of slavery were replaced with those of segregation and poverty. Though black people were no longer deemed three-fifths of a person, according to King, "the Negro...finds himself an exile in his own land," still unable to claim full citizenship.
He interestingly invokes both capitalism and revolutionary rhetoric in the second part of the speech:

In a sense we've come to our nation's capital to cash a check. When the architects of our republic wrote the magnificent words of the Constitution and the Declaration of Independence, they were signing a promissory note to which every American was to fall heir. This note was a promise that all men, yes, black men as well as white men, would be guaranteed the "unalienable Rights" of "Life, Liberty and the pursuit of Happiness." It is obvious today that America has defaulted on this promissory note, insofar as her citizens of color are concerned. Instead of honoring this sacred obligation, America has given the Negro people a bad check, a check which has come back marked "insufficient funds."
But we refuse to believe that the bank of justice is bankrupt. We refuse to believe that there are insufficient funds in the great vaults of opportunity of this nation. And so, we've come to cash this check, a check that will give us upon demand the riches of freedom and the security of justice.

King addresses the United States as a nation not only in possession of a wealth of financial capital -- which ought to be duly shared with the descendants of those who facilitated such immense wealth -- but also as a nation whose abundant freedoms and opportunities must also be shared.
He characterizes segregation as a "dark and desolate valley," echoing Psalm 23:4: "Even though I walk through the valley of the shadow of death, I will fear no evil...". He goes on to assert that justice belongs to "all of God's children." Here, Scripture is used to show racist white Christians that their racism is incompatible with their avowal of faith.
Toward the end of the speech, he uses "My Country 'Tis of Thee," a song that many people sang and continue to sing without contemplating its meaning, to show that, until inequality is dismantled, the words of this song will never ring true. Though the United States is a land in which the "fathers" of both black and white children have died, it is not for both a land of "sweet liberty."
Finally, he indirectly circles back to the Emancipation Proclamation with lines from a black spiritual, lines which have become the most memorable from the speech: "Free at last, free at last! Thank God Almighty, we are free at last!" With these lines, he suggests that the United States will only be free from hypocrisy when it allows black people to be free and full participants in citizenship.