John F Kennedy Jr would be the junior Senator from the state of New York today. He wins the seat in an easy, lopsided victory over the Republican nominee.
If JFK Jr had been running for the Democratic Senate nomination in 2000, I have doubts that Hillary would even challenge him. If she does, then JRK Jr still wins the nomination.
New York loves their celebrity candidates, and JFK Jr was even more of a celebrity in New York than was Hillary.
He would not be running for President in 2008, as he would not be ready for that step.
He would have endorsed Barack Obama in the 2008 Democratic Presidential nomination race.
JFK Jr would not be picked by Obama as his running mate. It might be seen as some kind of a pay off for receiving Ted Kennedy's endorsement. Besides that, JFK Jr would not be a particularly strong VP candidate. He would bring nothing in particular in the way of qualifications to the ticket besides his famous name.
He would very possibly run for the Democratic Presidential nomination at some point in the future, after he has matured somewhat, and after he has a solid legislative record under his belt.