Like Neike said, the Earth will be destroyed when the Sun goes bang-bang...
As for Humanity, there are two options, really. If we remain tied to Earth and do not overcome our problems with pollution and war and things, we will destroy ourselves many millions of years before the Sun swells up. If, on the other hand, we stop polluting and killing each other, and escape this planet, we could conceivably continue indefinately. Whether we would still be Humans once natural selection alters us to suit our new environments, however, is another matter...
If we do end up in space, I haven't a clue how we'll go extinct... I honestly can't think how, even though I know it will have to happen at some point...
BTW, having studied the theories, I believe more in the Big Crunch rather than Continual Expansion. After all, if there were little enough matter in the Universe for Continual Expansion, then Galaxies wouldn't hold together. Therefore there must be enough dark matter in the Universe for a Big Crunch...
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Guns don't kill people, People kill people! Using Guns.
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