How did Malcom X feel that black people should approach the struggle for equal rights?

Malcolm X did not agree with Martin Luther King Jr.'s concept of winning civil rights through nonviolent protest. Instead he argued:

Concerning nonviolence, it is criminal to teach a man not to defend himself when he is the constant victim of brutal attacks.

In other words, Malcolm believed that the violence and brutality with which whites had treated blacks over the centuries meant that blacks had a right to respond with violence. He believed that violence was the only language whites would understand.
Malcolm X, as a member of the Nation of Islam, also thought that integration was not the solution to black oppression. He didn't believe whites would ever accept blacks as equal. Instead, he referred to whites as "devils" and said that blacks must separate themselves from whites, cultivate pride, and build their own society.
Malcolm's views closely aligned with the beliefs of mainstream American culture which endorsed meeting violence with violence or hitting back the person who hit you. King's path was more novel to most Americans.
It's important to note that a pilgrimage to Mecca not long before his death began to radically change Malcolm's views. For the first time ever, he experienced whites and blacks treated as equals and began to consider the possibility of racial harmony and integration.