The ancient Nords used an Elder Scroll to cast Alduin forward in time. He re-appears shortly before the start of the game, and first reveals himself during the Helgen scene. He doesn't really have a specific reason to be there, but it is quite close to the Throat of the World, where he would have re-appeared. He may also have been able to tell that both a Dragonborn and a would-be Greybeard were there at the time.
Alduin has the power to raise slain dragons from the dead. Hence, the Dragonborn is the ultimate dragonslayer - he absorbs the souls of dragons, preventing them from being raised by Alduin. Note that the Dragonborn doesn't absorb Alduin's soul, suggesting he may be back again, one day.