I think his career stalls out in the House of Representatives, but he spends his entire life there and becomes incredibly well-respected, a la John Lewis. There's even the chance he becomes Speaker at some point.
A lot of the universal regard MLK has today is because he was cut down before his time. He would have been more controversial otherwise, especially with his growing anti-war activism and socialist leanings. He would have been a well-regarded civil rights leader no matter what, but he was likely too left-wing for a statewide election, most likely, to say nothing of the Presidency.
1. Anti-war leanings would have become irrelevant after the end of the Vietnam War.
2. Socialist leanings? He would have gladly sold out 'socialism' for equal rights for minorities. Of course he would have aligned himself with Big Labor.
3. If he fit the usual pattern, he would have been at his peak in his fifties -- in the early-to-middle 1980s, when Ronald Reagan was President. Could he have been elected to the US Senate?
4. How would he have dealt with the anti-communist movements of the Soviet Bloc? He was never a friend of the Soviet Union. KGB files on him show that the Soviet government had no use for him but saw much hope in black nationalist groups such as the Black Panthers.
