Humphrey and Johnson have a falling out in 1960, not sure what would cause it but the important thing is Johnson cannot stand Humphrey ITTL and vice versa. Assuming minimal butterflies, with Humphrey out of the running, McCarthy somehow winds up as the Vice Presidential nominee in 1964 in Humphrey's place. Because McCarthy cannot challenge Johnson in New Hampshire, Johnson continues his reelection efforts a little while longer, unfortunately for Johnson, the effort and stress of campaigning lead to his second heart attack, this time the result is fatal. Around February or March of 1968 President Lyndon Johnson is dead. McCarthy is sworn in as his successor. While President McCarthy attempts to reverse Johnson's Vietnam policies, his reputation among those opposed to the war is tarnished by his connection to the Johnson Administration. In the end McCarthy probably loses the election to Richard Nixon, though the election of 1968 will be quiet different with incumbent President McCarthy. McCarthy's Presidency is viewed mostly as a continuation of the Johnson Administration, and is among the shortest in American history, lasting slightly less than a year.
Yes I know the POD's vague, but I couldn't think of a specific POD in 1960 that would do the trick.