If there's a way to put a dent into the Kennedy machine, it has to be done early on. Let's say that for the sake of the challenge that HHH has a more organized and disciplined political organization backing him than OTL, and a better ground game than the Kennedy camp in Wisconsin. With a Humphrey win in Wisconsin, you probably do enough damage to shake up the Kennedy campaign, at least temporarily.
Then, let's say that Humphrey goes on to win in West Virginia with the help on an anti-Catholic smear campaign (likely not initiated by Humphrey himself, but cautious supporters of Humphrey, '527s' if you will). Kennedy sticks with the race, having achieved a few modest wins himself, and both campaigns show up at the convention with a good number of delegates, though Johnson is still in the lead. A few rounds of balloting leaves the convention in Johnson's favor, but without a majority for him, and the bosses are still a bit iffy on nominating a Southerner, but they aren't going to nominate Kennedy for damn sure, thanks to his perceived inability to carry a 'protestant' state like West Virginia.
The bosses eventually come to an understanding, and swing their support behind Humphrey, who nominates LBJ as his running mate to solidify the southern vote. Kennedy becomes a campaign surrogate for the Humphrey-Johnson ticket, and is widely mentioned as a member of a future HHH-LBJ administration.
On the Republican side, Nixon takes the GOP nomination rather easily, and chooses a midwestern Republican (Everett Dirksen) for his Veep in order to blunt Humphrey's edge in that area.
On election day, Humphrey carries the northeast and southern states, while Nixon takes the west and the midwest. Humphrey wins a small plurality in the popular vote over Nixon, as well as a bare majority in the electoral college.