What was the purpose of King's "Letter from Birmingham City Jail"?

When King wrote his "Letter from Birmingham Jail," the nonviolent mass protests of the Civil Rights movement were at their height and had gained national attention. While there was much vehement opposition to this movement and its goals, there was also a growing number of "sympathetic" people who argued that blacks did deserve more rights, but that it was not the right time. They criticized King's actions and called for him to wait for a better time. This type of argument was not new in American history.
King largely writes this letter in response to this argument. He poignantly points out that black Americans have been told to wait time and time again with little progress being made. He concludes that being told to wait really means "no" to their demands. He affirms his commitment to the Civil Rights movement and refuses to wait for a so-called "better time." Instead, he argues, black Americans deserve full and immediate political, economic, and social equality.

The letter was written by King in response to "A Call for Unity," a statement by a group of white clergymen from Alabama. In the statement the clergymen acknowledged the existence of widespread social injustice, but maintained that justice was best served through the courts. The clergymen were highly critical of King and his methods of waging the struggle for civil rights. They particularly objected to street demonstrations and protests, which they believed to have been whipped up by "outsiders."
King responds by saying that African Americans cannot simply wait for the courts to deliver civil rights; they must continue with their strategy of nonviolent resistance. In other words, justice delayed is justice denied. White Christians like the "Call for Unity" clergymen need to take a firm stand. The issue of civil rights is a moral question, one that has a fundamentally religious dimension.
In writing the letter, King attempted to emphasize the religious nature of the civil rights struggle. He does this in order to take the civil rights movement beyond the inherently contentious world of politics and law and into a realm of timeless truths and eternal justice. He's also encouraging those involved in the civil rights struggle to continue with the fight, to look ahead and see the day when they, and not those such as the clerics who wrote the "Call for Unity," will be hailed as heroes.