There are many metaphors in this speech, including the dream itself, the mountain, and the bad check.
When King says he has a dream, that dream is a metaphor. It is not something he was sleeping and thought up. It is a vision for the future of his country.
I have a dream that one day this nation will rise up and live out the true meaning of its creed. We hold these truths to be self-evident that all men are created equal.
In this case, King references the American dream of prosperity and financial independence. He also alludes to the Declaration of Independence. He is reminding his audience what it is supposed to mean to be an American.
Another example of a metaphor is King’s “bad check” metaphor.
Instead of honoring this sacred obligation, America has given its colored people a bad check, a check that has come back marked "insufficient funds."
King goes on to say that he refuses to believe that the Bank of Justice has gone bankrupt. We promised to make everyone in the country free, and did not keep that promise. He compares it to writing a check, a promissory note, agreeing to pay someone something and then not paying because you have no money.
One of the reasons the speech is so powerful is because of the strong imagery created through the metaphors. By using this colorful language, King helps us see what he is talking about, and touches an emotional chord. This helps make the speech persuasive.