Martin Luther King, Sr. was a preacher at the Greater Ebenezer Church and a tremendous influence upon his son's quest for racial justice. The cadence and forceful tones of King's Southern black preaching style greatly influenced his son's speaking style. For, King new that this style is one to which many relate and have faith in. Perhaps, then, you could analyze the effect of this preaching style with Biblical allusions, along with the rhetorical devices that King employs if this is the direction you wish to go.
If you are analyzing it for its literary merit, then the first two answers make sense. If, however, you are analyzing it from a historical and/or social sciences point of view, you might want to look at why King chose the ideas that he did and what impact the speech has had.
You might look at why he chose religious imagery as well as chosing to emphasize the equality of all people. You might then look at why it is that the speech went over so well and what the impact of that speech has been over the last almost 50 years.
There are a number of rhetorical devices you can look at if you are going to try and analyze the speech that way, including his incredible appeals to pathos (emotion) through the use of repetition and wonderful imagery at the close of the speech. If you are using a recording of the speech you can also look at the way he uses modulation towards the close to build tension and pressure almost like waves crashing on a beach building to a giant climax.
I think that more specific detail is going to be needed. Specifically, the question becomes what is being used as the standard to assess the speech. In my mind, I think that you could use several conditions to define the speech. On one hand, King's use of oratorical skills was on supreme display. Being able to seize the moment and use the historical condition of the Civil Rights Movement and the fight for equality for African- Americans to be able to speak in a manner that transcends historical condition would be a part of the speech's overall effectiveness. The use of repetition, as well as being able to invoke the spiritual elements through the use of the Bible in order to effectively convey Dr. King's point also makes the speech rhetorically effective. His ability to reach his audience, consisting of all Americans, with spiritual appeal and political appeals in terms of his use of American History also makes the speech incredibly effective as a work of rhetoric.