NOTE: It's pretty hard to "spoil" a true story everyone already knows about so I'll just give a quick rundown of what the film shows...
The film opens as we see a young Cassius Clay training for his upcoming fight against Sonny Liston. During his running we see clips of Liston destroying opponents in the ring and listening to Sam Cooke singing in nightclubs. Clay is outspoken, calls Liston many names (mostly about being big and ugly), makes up clever rhymes and doing whatever he can to win the crowd over. During all this prefight training, he also hangs out quite a bit with Malcom X and is becoming more and more interested in the Nation of Islam.
The fight finally starts and shows, almost in real time, Clay defeating the heavily favored Liston to become the Heavyweight Champion.
Clay meets and marries nightclub singer Sonji (played by Will Smith's real life wife Jada Pinkett Smith) but only after she converts to the Muslim faith.
He is also slowly becoming friends with Wide World Of Sports reporter Howard Cosell (Jon Voight) and the two of them, even though they don't show it on TV, have great admiration for each other.
At another fight he is so upset that Sonji dressed up for the fight, thinking it unproper for Muslim women to dress that way and ends up divorcing her. He remarries another girl whom he coincidentally met years earlier.
The FBI doesn't like the idea of this outspoken black man preaching the muslim faith and after Malcom X is killed and later Martin Luther King, Clay (now calling himself Muhammad Ali) is drafted.
The Muslim leaders want him to go ahead with the draft and just take a role in entertaining the troops and making public appearances but Ali would have none of that. He wasn't about to serve his country on principle and is sentenced to jail. He appeals the ruling but is stripped of his title and denied a boxing license in just about every state. He is broke and the Nation of Islam will no longer allow him to represent them. Despite all this, He claimed "I never stopped being a Muslim, I never stopped being a champion."
Finally the supreme court ruled in his favor 8-0 and he is free again but has lost the 3 prime boxing years of his life.
One of his trainers, Bundini Brown, is now broke and actually sold Ali's championship belt for drugs, pleads with Ali to take him back which he does.
He prepares his comeback by defeating Jerry Quarry which sets up the "Fight of the Century" with "Smokin" Joe Frazier. In a 15 round decision, Frazier defeats Ali.
The final 45 minutes of the film show Ali preparing for the "Rumble in the Jungle" with George Foreman. This is the first time we see Don King who is the promoter of the fight.
Ali is a hero in Zaire, Africa and is greeted with chants of "Ali Buma Ye" (Ali, Kill Him) wherever he goes.
The fight is delayed for a few weeks due to a Foreman training accident and Ali's wife goes home. Ali meets a beautiful reporter and a relationship begins. When his wife returns she is furious with him.
The fight finally begins and Ali looks horrible using the "rope a dope" technique he suddenly came up with. He just sits back on the ropes allowing Foreman to take many swings at him but having no effect. Bundini is practicallyy in tears, pleading with Ali to get off the ropes. After a few rounds of this, Foreman is worn out and the 32 year old Ali defeats the 24 year old Foreman.
The movie ends with an image of Ali standing on the ropes, cheering with the crowd after his victory.
Just before the credits we learn that Ali divorced his wife, married the reporter who he later divorced and married another woman who he now lives with in Michigan.