Hmm... Well, let's say RFK isn't assassinated in '68 - that's our first step.
In the rest of the primaries, RFK narrowly loses each one, and does not win the Democratic nomination - he loses to HHH. HHH loses to Nixon, as per OTL, and we have a Nixon presidency. Without butterflies, we'll say Agnew still resigns, as does Nixon, and Gerald Ford becomes 38th President and runs in '76. Something happens - I can't think of what, unfortunately - where RFK does not run for president, and Jimmy Carter still wins the Democratic nomination and beats Gerry Ford. Basically, this part is basically relying on RFK not changing politics completely, which may be ASB, but this is just a rough draft idea here.
In 1980, with the Carter presidency failing, RFK challenges Jimmy Carter for the Democratic nomination, and knowing RFK and his charm and Carter's unpopularity, RFK manages to unseat President Carter, and easily, Ronald Reagan is the Republican nominee. Voila, an RFK vs. RR match in 1980, with no incumbents. This part is completely plausible - but the two problems with this scenario is: 1) How the devil does RFK not run for president in 1976? and 2) How the devil do we get him not to majorly affect the years 1968-1979?
EDIT
Alright, I just got an awesome idea.
First off, Richard Nixon wins the gubernatorial election of 1962 in California. However, the Watts Riots of 1965 turn out horribly under Governor Nixon, and Nixon does something to lose the support of many Californians (not too hard, really). In 1966, Governor Nixon, with a 35%-or-lower approval rating in California, realizes he cannot win re-election as Governor and declines to run. Ronald Reagan campaigns well and captures the GOP nomination and endorsement for Governor that year, and beats whoever the Democrats nominate and endorse. Thus, Richard Nixon is out of the political arena for '68, and Reagan is in.
It probably won't be too hard for Reagan to win the Republican nomination in 1968, but it won't be easy either. But he does, for scenario's sake. RFK also is able to beat LBJ and Eugene McCarthy in the Democratic primaries and narrowly wins the Democratic nomination. Thus, you have an RFK vs. Ronald Reagan campaign in 1968.