"I had to prove you could be a new kind of black man," Ali told me. "I had to show that to the world." Given the time period and the country where he lived, what did Ali mean by this quote?

In this quotation Ali is referring to the early 1960's, when the civil rights movement in America was gathering momentum in the face of entrenched and institutionalized racism. In the southern states of America the Jim Crow laws still legalized racial segregation. In 1963, between 200,000 and 300,000 Americans marched on Washington to demand equal civil and economic rights for African-Americans. Amidst this unrest, in 1964, Cassius Clay changed his name to Muhammad Ali, and in so doing began to forge a new identity as a black man in America. Cassius Clay was Muhammad Ali's "slave name," and so in changing his name he was symbolically casting off the shackles of slavery which bound him to that past. He was forging his own identity as an independent man, and the first step was to cast off his slave name and take instead what he called his "free name." This, in part, is what Ali meant when he said that he "had to prove you could be a new kind of black man."
Ali is also referring in this quotation to the representation of black men in boxing in the early 1960s. The three most prominent heavyweights at this time were Floyd Patterson, Sonny Liston, and Muhammad Ali. Floyd Patterson was gentlemanly, softly spoken and was cast by many as "the good negro." He was an advocate of the civil rights movement but at the same time didn't advocate too loudly. Sonny Liston, on the other hand, was cast as "the bad negro." Liston had a criminal record and also had ties with the mob. He was portrayed as animalistic, monstrous, and threatening. He was, in short, cast as a personification of every lazy, racist stereotype of black men. When Ali said that he wanted to be a "new kind of black man," he in part meant that he wanted to break free of the stereotypes that Patterson and Liston had been saddled with.
It's reasonable to argue that Ali was successful in forging for himself a new identity as "a new kind of black man." He was neither deferential like Patterson, nor monstrous like Liston. In 1968, he demonstrated his independence when he refused to be drafted during the Vietnam War, declaring that, "I will not disgrace my religion, my people or myself by becoming a tool to enslave those who are fighting for their own justice, freedom and equality." Although at the time Ali was branded by many as a traitor, his stance on the Vietnam War has since been vindicated, and is now widely perceived as brave and virtuous. He is now, almost three years after his death, regarded as one of the most unique, most loved and most respected figures in all of America's sporting history.