It really depends when Clinton dies in office.
Lets say that Clinton dies before 1998, there is no impeachment proceedings.
If he dies say in 1997 shortly after taking his 2nd term then Gore has about 3 years. The bad thing for Gore is he is facing Republican majority in both chambers in 1997 and 1998. In 1998 elections the Senate will become equally split.
Gore will continue in state of the unions to call for more green jobs and talk about global warming. His ideas will not pass the Congress.
I believe of the choices he'd choose Ann Richards, former Gov. of Texas as his VP, and I do believe that the Congress would go along with it. (She's far better than many choices Gore could have chosen, in fact he might first nominate a more leftest VP, have Congress reject it and offer Ann as a compromise candidate).
This means in 1999 and 2000 the deciding vote would be Ann Richards in the Senate. Ann's personality also picks up the ticket a bit, counteracting Gore's wooden personality.
Other than his call for green energy I don't really believe anything substantive could change. In 2000 Gore/Richards would run for reelection and I do believe they would win. Gore has served more than 2 years of Clinton's last term so he cannot run again.
He's still dealing with an evenly split senate for the next two years and Republican majorities in the House. He will not be able to pass his green agenda before the 2002 midterms. After the 2002 midterms Senate has a Democrat majority and the House is still Republican.
9/11/01 will not be prevented. It's almost a fixed point in time. Gore's response will probably be much more tepid than that of Bush's. There is no Patriot Act passed in the coming month, if it passes at all it will be months down the road, and probably much weaker than the Bush one.
Gore would concentrate on the Airlines and lack of secruity, however he would probably fall back on legislation that had failed in the past to upgrade the protection of flight crews with changes to the hatch to the crew compartment. He probably would have put marshall's on planes too. Ultimately though different than Bush's angle this still prevents anymore planes from being used after 9/11. No Dept of Homeland Security is formed and no TSA is formed.
Gore with Richards would probably pass a stronger Border defense package beefing up border security on both borders, increased checkpoints, flyovers etc, no wall. (I believe he'd double the border security).
They would push for a new immigration bill and ultimately under Gore I believe such would pass in the fading days of his only full term, with Richards leading support in the Senate and many Republicans in key districts deciding to vote for it. The bill would be similar to McCain-Kennedy.
In response to the Beltway sniper they call for greater gun control, but mostly goes on deaf ears in Congress, though some rules for assault weapons etc pass even though those weren't used in the sniper shootings.
In 2004 Gore is ineligible to run again. Ann Richards would run for president (she is not the first woman to do so by far), and she picks Howard Dean as her Vice President. She's the first woman to be on a major party ticket, she wins the election not decisively but well enough. Federal Defense of Marriage Act does not pass (or if it does is vetoed).
Unfortunately Ann is found to have cancer in March 2006, her treatments will leave her unable to fulfill her duties for many months, thus in May of 2006 she resigns the presidency letting Howard Dean take her place, she would die in September. She would be given a state funeral. Wesley Clark is tapped to be Dean's Vice President and is confirmed by Congress.
Dean/Clark will see the House flip in the 2006 elections to the Democrats. The Senate remains tied but with the 2 independents who vote with the Democrats they have a majority. Dean while popular with the left wing of his party is not as popular with the American people, he gets a carbon tax bill through the House, but is unable to get it through the Senate. He ends Don't Tell, Don't Ask.
Dean faces the same question Gore did, he has served more than 2 years of Richards term, he can only thus serve as President 1 more time. The last part of his term though faces the housing bubble, the failure of GM and Chrysler, and a huge banking crisis. A bailout of the banking industry passes, with broad Democrat and Republican support for fear of a total financial collapse of the United States. It is unpopular even though President Dean supports it. Bailouts for GM and Chrysler are on the table as the election comes. Dean/Clark still run for election though.
Americans vote with their feet. Tired of Democrat administrations that have led them down this road, they instead choose to vote for the Republican in 2008, John S. McCain becomes president of the United States. (How the people thought this would be better after his support of the stimulus is anyones guess). His VP choice to shore up his minority and conservative credentials is Alan Keyes. For the first time in many generations the Democrats loose the black vote and the hispanic vote. The election is a stunning defeat.
McCain still supports the bailout of GM/Chrysler but his VP is opposed to such and it makes no headway in Congress. GM and Chrysler are forced into Bankruptcy Court. Ford enjoys a big boon and actually acquires some of the GM/Chrysler plants from Bankrupty sales, expanding its production in the United States. A couple of smaller companies emerge from the bankruptcy as well while GM survives as a smaller company, Chrysler is lost. The initial shockwave sends the Dow Jones plummeting, as it is clear now there will be no more bailouts. However after about the initial 6 months the Dow Jones regains and actually advances.
Congress does come up with a stimulus package, however rather than the OTL it more resembles FDR's Works Progress program of the 1940s. Concentrating on road, schools, bridges and other critical infrastructure, including an overhaul of the nation's electrical grid. Democrats loathe to oppose something essentially passed by FDR go along with the package. This package though is more popular with the American people than OTL.
In addition an educational bill passes giving assistance to low and middle income families to pay for technical schools and college.
In Dec 2009 the next terrorist attack occurs, A man having explosives in his underwear destroys a plane over Detroit, debris from the plane, jet fuel etc lead to extensive damage of Detroit. A firestorm consumes about a quarter of the city before its contained, thousands are presumed dead in the following aftermath. Its even bigger than 9/11 as far as disasters go.
McCain touring the devastation calls for Federal Disaster aid to Detroit and Michigan and congress quickly responds. Keyes calls for greater border security and more air marshalls. Northwest Airlines is sued by Detroit citizens over the lack of security on their planes. The airline would be forced into bankruptcy by mid 2010 as lawmakers after the GM/Chrysler outroar that led to the 2008 elections are not willing to bail it out or give it immunity.
The United States enters Yemen with the goal of cleansing Al-Queda in the Arabian Peninsula, working in concert with the Yemeni government. Using special forces, drones and other modern gear they quickly root out several of the main safe houses of Al-Queda and deal a blow to it in Yemen by 2011. It is clear though to help maintain order US Troops will have to remain in the area.
In 2012 Hillary Clinton wins the nomination, picking Bill Richardson as her VP. The outpouring for support of the widow easily wins her the nomination in the primaries.
This sets up Clinton/Richardson vs. McCain/Keyes.