There were a multitude of factors which culminated to cause the sack of Constantinople. I'm adapting this answer from a previous post of mine, which mostly explains the political aspect of the sack.

Overall, the general image of the Eastern Empire in the West was not a very good one; but religious and cultural differences cannot be said to have been the only driver in the sack. Rather, it has much more to do with the fact that fourth crusade was fundamentally run by lesser princes and nobles, and rather haphazardly at that. And importantly, the government of the Republic of Venice headed by the wily doge Enrico Dandolo wasn't as enthusiastic as it had been about previous crusades.

During the third crusade, the Doge Mastropietro had carefully assessed the wealth of all leading families and decreed a forcible loans to be made to the state in order to construct ships to be put at the disposal of the crusading armies. But decades later, although Venice was again picked by the crusade's leaders as the starting point of the expedition, the Venetians would not put up the money to outfit the ships necessary. The Holy Roman Emperor, Frederick Barbarossa, had only a few decades prior concluded a series of devastating wars in Northern Italy, and while the Venetians had tried as best they could remain outside the conflict, the war's instability had hurt Venetian trade, and subsidies to Barbarossa's opponents had depleted the coffers of the state and strained taxpayers.

A compromise was reached: the crusaders would first stop in the Dalmatian city of Zara, a Venetian client which had recently rebelled against the Republic's authority, fight to re-instate Venetian rule, and in return the Venetians would ship them to their final destination.

The reaction to this first deviation was not positive: Pope Innocent was furious, and excommunicated the Venetians for having rerouted a crusading army for their own purposes. However, this had little effect on the crusaders, or the rest of Europe for that matter! Although certainly angry at the Venetians and the crusaders, at the time of the fourth crusade Western European leaders were particularly preoccupied with other matters: The King of Sicily was a five year old boy in the care of Pope Innocent III, while in Germany Otto of Brunswick was in open war against Philip of Swabia for the title of Kaiser; and although France was ruled by the energetic Philip Augustus, the country was split between the Capetians and the Angevins, and the latter were coping with the death of Richard Lionheart. In fact, Richard's successor John's epithet "Lackland" didn't exactly promise effective rule. Lastly, in the east Emperor Isaac II had been dethroned by his brother after ten years of internal conflict, and to garner the support of the nobility the new Emperor had extended favors and privileges compromising his ability to rule.

A further change of plan took place once the crusading army was camped outside Zara and the city had surrendered. Alexius, the son of the deposed Byzantine Emperor Isaac, had managed to escape from his prison and sought refuge at the court of his cousin, the aforementioned Philip of Swabia. Philip, probably to strengthen his aura of authority as he fought to claim the german crown, took up the guise of a seasoned statesman and sent emissaries to propose a further redirection of the crusade: if the expedition would instate Alexius in Constantinople, the young pretender would commit to bankroll the remainder of the expedition and provide ten thousand additional soldiers, in addition to paying for five hundred knights to remain in the Holy Land after the conquest. An offer too good to pass up!

Of course, once Constantinople was taken and Alexius placed on the throne, the depleted state of the Byzantine coffers made keeping such promises impossible, leading the crusaders to conclude that sacking the city was the best way to obtain that which was owed them. The Venetians under Dandolo, the natural intermediaries, didn't do much to encourage cooperation and even refused to set sail for the Holy Land until Alexius fulfilled the promises he couldn't keep.

A King or Emperor at the head of the Crusade might have been able to find alternative courses of action. Or maybe not: Venetian relations with the Byzantine Empire were at their absolute lowest point, and a series of escalating disputes in prior decades had even led the Venetian consul in Constantinople to be jailed. In fact, the jailing of the Venetian consul prompted the doge at the time, Vitale Michiel, to organize an enormous punitive military expedition against Byzantium. But the Emperor Manuel Komnenos outwitted him, cleverly setting up circuitous peace talks as the Venetian army and navy was worn down by attrition in the Aegean Sea. When the disease-ridden army returned to Venice, Doge Vitale Michiel was lynched.

The consul jailed by Komnenos was none other than future doge Enrico Dandolo. With many of the mutineers against Michiel now risen to leadership positions, there was very little sympathy in the Venetian camp for the Byzantine predicament. The Crusade's leaders thus were under immense pressure from the Venetian leadership to take an uncompromising and punitive stance.

As I implied above, is possible that the presence of a more resource-rich leadership, or at least a more authoritative one, might have prompted a different course of events. So had the Venetians been paid in full, you could argue they probably would have shipped off the crusaders to their destination and only tag along in small numbers to negotiate trade agreements (as they had done in prior crusades). As futile as it is to discuss a counterfactual, maybe a King or Emperor would have been able to persuade the Venetians to accept as payment larger concessions in the holy land; after all, the leader of the expedition was one Boniface of Monferrat, who Dandolo was able to undermine at every turn. After the sack, Dandolo was even able to single-handedly elevate Baldwin of Flanders over Boniface as the new Emperor.

But as the crusading bug filtered down from the great monarchs of Europe to the mid-and-lower tiers of the aristocracy, who in France and Italy had especially benefitted from conflicts which by the time of the fourth crusade had resulted in a sort of stable fragmentation of political authority, it might seem natural that a more stable organized polity would take advantage. Although it's also true that another lens might paint a different picture; I'm writing based on having read a lot of material from the Venetian perspective.