@The Wizard:
Partially right. The crusaders had with them one Alexius Angelus, whose father was emperor Isaac II. Isaac again was deposed and blinded by his brother Alexius (became Emperor Alexius III), and both Isaac & his son Alexius were put in prison.
Alexius, the younger fled in 1201, and somehow met up with the crusaders and asked for help. Crusaders agreed and journeyed to Constaninople. There they demanded Alexius III step down and give crown to Alexius Angelus. That didnt happen, so they laid siege to the town. During the siege, Alexius III fled and as I recall only the Galata-section of the town had fallen to the crusaders.
Alexius Angelus was put on the throne as Alexius IV, and his father Isaac II was made co-emperor (Greeks didnt trust Alexius IV). To cut a story short, Alexius IV owned the crusaders money, but didnt have enough money to pay. So the crusaders attacked again, but a Byzantine noble called Murzuphlus seized the throne and had the two emperors killed, and made himself Alexius V. And during this time, Constantinople did fall, but even though the Greeks could have been able to continue fighting Murzuphlus fled as well. And then Count Baldwin of Flanders was placed on the throne.
For a more detailed explanation:
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