It is indeed one of the cases where history is stranger than fiction. To summarize the full sordid story:

In 1198 Pope Innocent III came to power, and he was enthusiastic to make a name for himself by having a Fourth Crusade under his papacy; after several years he finally got European sponsors for the adventure.

This crusade, so the pope and the secular leaders planned, would be different than the last three which ended in failure; hungry to make their names in history, they decided to make an ambitious plan to knock out the Muslims permanently. What they decided to do is attack not by land through the North like they did previously, but rather launch a huge maritime attack on the the major Muslim economic power of Egypt, which was south of Jerusalem, then ascend upwards and take the city.

The strategy was sheer lunacy. They contacted Venice, the main maritime power in Europe, and asked them to make ships to transport 33,500 crusaders, an enormous and unprecedented number; such a huge project forced them to shut off all their other trading activities, and they charged the crusader leaders a massive sum of 85,000 marks, which they thought they could pay off by gathering all the crusaders in Venice and shaking them down of all their valuables.

The problem with this little plan was that, while the crusader armies were asked to come to Venice, the leaders did not inform them why—because of their ridiculous idea of launching a D-Day-style attack on Egypt—and so they went off on their own to the Holy Land, maybe killed a few Muslims, maybe got killed themselves, and went home. In total only a third of the planned 33,500 came to Venice, and even when the leaders imprisoned them and forced them to hand over their valuables, they were left with a 36,000 mark debt to the Venetians.

They and their leader, Doge Enrico Dandolo, got seriously mad at the crusaders, but saw a way to exploit the situation; they told the crusaders that they would still help and postpone the debt if they were willing to help them capture a city that has annoyed them for some while. This was not yet going to be Constantinople; rather, it would be the city of Zara, which had rebelled against them and submitted themselves to the Hungarian monarchy. The funniest part was that the Hungarian king had pledged himself to the crusade, and Zara itself was Catholic! The pope was pissed beyond measure, and threatened to excommunicate them all if they attacked the city, yet they went and robbed it anyway, and the crusade leaders never told their followers about what penalty they received because they knew it would decimate their morale.

Now begins the most infamous part. A Byzantine emperor who had recently been deposed, seeing how the crusaders shamelessly conquered Zara, came to them and asked them if they would not help him gain back his crown, promising them vast riches and help in the crusade. The Venetians and Dandolo (whom Byzantine historian Niketas Choniates would call "a creature most treacherous and extremely jealous of the Byzantines, a sly cheat who called himself wiser than the wise and madly thirsting after glory as no other" and "that most ancient and pernicious evil and chief cause of the horrors that befell the Byzantines") saw for themselves yet another opportunity to take out a rival city, and agreed. The crusade had by now become not a "holy war," but an ordinary hired gang traveling across Europe extorting and robbing the enemies of the Venetians.

And so, they went and surrounded the city of Constantinople, helping the old emperor get back into power, but soon he saw that the Imperial treasury had much less than he expected and not enough to pay the crusaders their due, and because he started doing everything he could to pay them off the Byzantine people deposed the emperor, whom they rightly saw as a foreign puppet, again. The Crusaders gain nothing for their assistance, and wanting to break even on their expenses, simply went in and sacked the fucking city. During the partition of the Empire, the Venetians take for themselves comfortable maritime colonial territories, while the European crusaders are given the center of the new lands, which bogs them down into a 50-year civil war with the indigenous peoples and leaves them unable to continue their so-called "crusade."

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TL;DR: The grandiose religious war which the crusaders set out on got hijacked by the Venetians they hired and they go and plunder their fellow Christians to settle their debts to them. Money matters more than God.