If you are looking for some examples of hyperbole in King's "Letter," here are some examples:
The nations of Asia and Africa are moving with jetlike speed toward the goal of political independence, and we still creep at horse-and-buggy pace toward the gaining of a cup of coffee at a lunch counter.
In this comparison, the difference in the speeds at which Asia and Africa are moving toward civil rights compared to the U.S. is so vast, that one can consider it to be an exaggeration.
Another example comes in the same paragraph:
There comes a time when the cup of endurance runs over, and men are no longer willing to be plunged into the abyss of despair.
One will again notice that King's descriptive language makes this statement a hyperbole. While "despair" is already a negatively connoted word, the "abyss of despair" is the very depths of the negativity, for an abyss is a seemingly bottomless pit.
Further in the letter when King is discussing just and unjust laws, he uses the example of Adolf Hitler and the Hungarian freedom fighters, who revolted against the Soviet-controlled country:
We should never forget that everything Adolf Hitler did in Germany was "legal" and everything the Hungarian freedom fighters did in Hungary was "illegal." It was "illegal" to aid and comfort a Jew in Hitler's Germany. Even so, I am sure that, had I lived in Germany at the time, I would have aided and comforted my Jewish brothers. If today I lived in a Communist country where certain principles dear to the Christian faith are suppressed, I would openly advocate disobeying that country's antireligious laws.
Using such extreme examples could be considered hyperbole, but these examples are significantly effective in proving King's argument that people should fight against unjust laws, even if those laws are considered legal at the time.
Overall, one would not consider his letter to be hyperbolic, but King's use of hyperbole is found in his descriptive language and the examples that he uses.