Explain the relationship between O. J. Simpson and civil rights movement during his career at USC. What are other black athletes doing to promote civil rights? What is O. J. doing? What does O. J. think he needs to do to be successful in America?

I strongly recommend that you view Ezra Edelman's film, O. J.: Made in America. Edelman explains Simpson's relationship to the civil rights and black power movements and depicts why Simpson maintained distance from both movements during his time as the star of USC's football team.
I would describe O. J. Simpson's relationship to civil rights and black power as distant and uninvolved. To maintain his status within USC, which was a rather conservative institution, he tried to come off as non-threatening. Taking a political stance, as we've seen recently, is risky for athletes, particularly black athletes who often face additional pressure to conform to social expectations or risk being rejected by the public.
Of course, there were others who took the risk of losing their careers in favor of standing up for their principles. Muhammad Ali is the best-known of these athletes. He faced social condemnation for his refusal to go to Vietnam after being drafted. Other athletes who expressed their support of civil rights and black power are Tommie Smith and John Carlos, who won the gold and the bronze respectively for track and field at the 1968 Summer Olympics, and who famously gave the black power salute.
While Ali was known for his vocal opposition to the Vietnam War and his vociferous support of the Nation of Islam, to which he belonged, Smith and Carlos's gesture demonstrated silent but powerful support. A contemporary parallel would be Colin Kaepernick's silent gesture of taking a knee during the national anthem.