We're not quite sure what happened to Constantine XI, only that he was most likely dead when the siege was over. According to Michael Critobulus, who was a historian working for Mehmed II during the siege, Constantine XI supposedly tore off his imperial regalia and participated in the last charge of his troops against the Ottomans who had broken through the walls, where he died. To me, this is the most likely account, although there are accounts that also say that he escaped (Tursun Beg, who was a Turkish officer), and also one where he was trampled to death running away (Leonardo of Chios). We can't be sure, really, but he most likely died in the fighting. The body that people thought was Constantine XI was beheaded and given to Mehmed and then paraded around the city (supposedly identified by eagles embroidered on his socks), but according to the sources this body didn't cause the citizens to respond, so it might not have been him.

Or you believe the myths and hype and he was turned into a marble statue and is buried under the Golden Gate, waiting for the Greeks to rise and reclaim their city, where he will come back up and rule the empire once more.

I don't know the fate of the emperor but since Constantinople was taken by storm, almost the entire population had been captured as prisoners of war and their properties taken as booty. But Galata (Pera), on the other side of the Golden Horn, had surrendered under ahdname, and consequently the population was spared a similar fate.

As for what happened to the Greeks after the conquest: Sultan Mehmed's main goal was to make his capitol a thriving center of trade. He showed great efforts to repopulate and rebuild the city. Surveys ordered by the Sultan in 1455 and 1477 show that after the Muslim majority, Greeks were the largest minority in the capitol. They held important positions in the governance and trade. Members of the old Byzantine aristocracy, the Palaiologoi, the Kantakouzenoi, the Chalkokondylai, and the Raouls were prominent taxfarmers under Mehmed the Conqueror and his successors.

Many more details can be found in Halil İnalcık's "Greeks in the Ottoman Economy and Finances 1453-1500", 1993 which can be dowloaded for free from here