Why was King such a controversial figure in the 1950s and '60s?

Martin Luther King was a controversial figure for a number of reasons, but mainly because of his commitment to unlawful, yet nonviolent, resistance. Many of those normally sympathetic to the cause of civil rights criticized King for staging unlawful protests and demonstrations against racial oppression and its legal foundations. They argued that the civil rights struggle should be conducted purely through the courts, not on the streets or in segregated lunch counters.
It was this criticism, voiced by a group of white Southern clergymen, that inspired King to respond with his famous "Letter from Birmingham City Jail," in which he justified the civil rights movement's approach to securing justice and racial equality through unlawful means. The courts were part of an apparatus of racial oppression and couldn't therefore be relied upon to secure civil rights. That being the case, it was necessary to resort to the kind of direct action seen on the streets of Birmingham and other cities throughout the Deep South.
From another direction, King was criticized by more militant black civil rights campaigners, who argued that some degree of violence was not just unavoidable, but necessary to advance their cause. They claimed that passive resistance was all very well, but unless African Americans actually rose up in anger, using whatever means necessary to gain equality, then they wouldn't be taken seriously by the powers that be and would remain trapped in a state of oppression. For such radical voices, King's approach was too riddled with compromise to be embraced; they believed direct action needed to be combined with violence to make white society sit up and take notice.