Ted Kennedy's best shot at winning the presidency was in 1976. The reasons he didn't run that year was because of his personal family problems (Joan Kennedy's alcoholism and drunk driving arrest in 1974) and because of the media's returned interest in Chappaquiddick. Also his lack of real desire to be president might have also played a role in him sitting out in 76. But even with all that Kennedy still led the Democratic field for the 1976 race in a Harris poll conducted in April of 1975.
If their is no Chappaquiddick and Mari Jo Kopechne lives, despite his family problems Kennedy might run in 1976. Joan stood by him and campaigned for him in OTL in 1980. I think she would have done the same in TTL in 1976.
He could have gotten many of the uncommitted votes in the Iowa caucus and beaten the unknown Jimmy Carter in Iowa or came in a close second to Carter. The next three primaries were in New Hampshire, Massachusetts, and Vermont. I certainly think EMK would have won those state's primaries handily. After Kennedy wins the Wisconsin primary on April 6th, I see Carter dropping out of the race unable to raise money or gain any real traction. But he would have came in second to Kennedy in many of the primaries possibly even winning the Florida and North Carolina primaries before dropping out. Kennedy would have had more than enough delegates to win the nomination months before the convention, while Ford and Reagan would have been battling it out all the way to the Republican Convention.
Kennedy's first choices to be his running mate would have been fellow senators Birch Bayh, John Culver, and Walter Mondale. But his senior campaign adviser Ted Sorensen, campaign manager/brother-in-law Stephen Smith and DNC Chairman Robert Strauss would have convinced him to pick Carter to be his running mate. Carter would have balanced the ticket with Carter bring a southerner, a protestant, a WWII vet, working class background, a former governor, and a moderate. None of which was Kennedy.
I think Kennedy would have beaten Ford in the general election by a comfortable margin.
Gerald Ford/Bob Dole - 184 electoral votes
Ted Kennedy/Jimmy Carter - 354 electoral votes
EDIT: I'll try to finish this timeline and talk about a EMK presidency starting in 1977 sometime later this week.