Nixon decides against the 'Southern Strategy' in 1968, instead running on a platform not too different from LBJ's of four years prior. He promises stronger federal action on Civil Rights, environmental protection, and a reform of the Great Society to make it more efficient (i.e. his 'FAP' plan), while also criticizing the Johnson administration for it's failure to win in Vietnam.
OTL, Clinton (ATL Rodham) left the GOP because of it's abandonment of the principles that Rocky and the Eastern Establishment stood for. I'm assuming, that with someone like Nixon, a 'Western transplant' running as an 'Eastern Establishment' candidate, she might stick around. So let's say, for instance, that she does. She goes on to Yale Law School as per OTL, but her political preferences clash with a certain William Clinton, and the two thus don't end up dating in 1971.
The Nixon administration is nearly the same as OTL, minus some of it's more conservative rhetoric. The Democrats nominate Hubert Humphrey again in 1972 (likely a result of butterflies and Democratic criticism of a more pronounced detente), who loses by a good margin to Nixon, but not to the degree of OTL McGovern's loss. As a result, Nixon does a lot more 'rat fucking' to try and discredit Humphrey, resulting in the Watergate burglaries in a similar fashion to OTL.
Rodham lands a job with the Office of White House Counsel following Nixon's re-election, and serves in that capacity until Nixon is impeached in 1974, resigning prior to the impeachment in protests of Nixon's actions. President Ford asks Rodham to stay on, but she refuses, but does accept an appointment as a U.S. Attorney in Chicago in late 1975. When Ford lost his own bid for re-election to Democrat Henry Jackson of Washington, Rodham planned to tender her own resignation, but was reappointed to her post by President Jackson, who had a great deal of respect for Rodham for her actions during the Nixon impeachment.
Rodham stayed on in this capacity until 1980, when she sought the Illinois Attorney General spot when a vacancy became open. In her first campaign, Rodham ran as a social liberal and a fiscal conservative, willingly nominating herself as a 'Rockefeller Republican'. She won, despite the re-election of Democratic President Henry Jackson that same year, carrying the office by a modest margin.
In 1984, she campaigned for the Republican John Connally in the general election, despite her personal distaste for his social views. Connally was defeated by the incumbent, President Carter (Jackson had passed away in 1983). She supported the campaign of fellow Illinois Republican Donald Rumsfeld in 1988, and was offered a spot in his administration following his victory over President Carter, but she declined, seeking to move up in Illinois politics herself. Rodham was elected Governor of Illinois in 1990.
From 1991 to 2007, Rodham served as Governor of Illinois, governing as a Rockefeller Republican. She increased education funding, pushed through welfare reform measures, and cut spending overall. Her approval ratings remained high throughout her tenure as Governor, which ended when she decided against seeking a fifth term as Governor in 2006. Rodham then focused on preparing herself for a Presidential campaign, which seemed to be tailored fit for a Republican victory with the unpopularity of the Lieberman administration.
Facing down former Governor Mitt Romney in the primary, Rodham would emerge after a contentious primary fight with that would last until the spring. Rodham officially became the presumptive Republican nominee in March of 2008. She would get a head start over Democratic nominee John McCain, who was unable to fight off primary opponents to his left in the early stages of the Democratic primary. Nevertheless, Rodham would go on to choose former Governor Bob Erlich of Maryland as her running mate, to face down the Democratic ticket of Senator McCain and Senator Barack Obama of Illinois.
The resulting campaign proved bloody between the McCain and Rodham camps, with the McCain camp accusing Rodham of being 'soft on Communism', while Rodham tied McCain to the unpopular Lieberman administration and the ongoing conflict in North Korea. In the end, history was made with the landslide victory of Hillary Diane Rodham, who would become the first woman elected to the presidency in American history.
In the months since her inauguration, President Rodham has dealt with a mild economic downturn, reforming the nation's Medicare program (public option under Nixon, full expansion to a single-payer system under Jackson-Carter), the growth of Soviet influence in Iraq and Afghanistan, and forming a cohesive strategy for the North Korean conflict. As of this writing, her approval rating stands at 55%.