The Russians were quite interested in the events of the crusade. They weren’t too interested in the specific doctrinal differences between the Greek and Roman churches, but they knew it had something to do with the other crusades that targeted Jerusalem. The crusades in general are mentioned relatively rarely in Russian sources. They were aware of the First Crusade and happily reported when Jerusalem was conquered in 1099. Russians must have already been going on pilgrimage to Jerusalem long before the crusades, as did Christians from elsewhere in Europe, and this tradition continued under the crusaders - in fact one of the earliest pilgrimage accounts of the crusader kingdom is from a Russian monk, Daniel of Kiev.

Constantinople was much closer to home though, and was the spiritual capital of their own church. There was no division between the Greek and Russian Orthodox churches yet. Russia had been converted by Greek missionaries from Constantinople and the various Russian states remained closely linked with the Greek church and the empire.

One Russian chronicle, the “Chronicle of Novgorod”, is pretty well-informed and includes several pages about the events of the crusade. It mentions how Emperor Isaac II (Isak) was overthrown, blinded, and imprisoned by his brother, Alexios III (Oleksa), and how Isaac’s son, also named Alexios (Oleksa Isakovits), escaped from Constantinople and met with the German “tsar” Philip of Swabia. Philip and the Pope then organized a military expedition to restore Isaac to the throne.

“But the Franks [i.e. Latin Europeans] and all their Voyevodas conceived lust for the gold and silver, which Isakovits promised them, and forgot the commands of Tsar and Pope.” (Chronicle of Novgorod, pg. 44)

There’s an abbreviated account of the siege - the crusaders attacked and there was a fire in the city; Alexios III abdicated and restored Isaac; Isaac was then overthrown again by his son who became Alexios IV; but the people of Constantinople rejected Alexios IV and he was overthrown by yet another Alexios, Alexios V Mourtzouphlos (Murchufl). The crusaders were waiting for Alexios IV to reward them for their help, but clearly that wasn’t going to happen now, so they attacked Constantinople again and took it in April 1204.

“And so perished the empire of the God-protected Kostyantingrad and the Greek land in the quarrel of Tsars; and the Franks rule it.” (Chronicle of Novgorod, pg. 48)

The chronicle knows the main leaders of the crusade, although by their titles rather than by name: “Markos”, i.e. Boniface, the marquess of Montferrat; “Kondoflandr”, or Baldwin, the count of Flanders; and “Duzh”, or Enrico Dandolo, the doge of Venice. The Russians know all about Dandolo’s blindness and the story that he was blinded on purpose by the emperor Manuel. All of this information must have come from an eyewitness - the chronicle goes into great detail about the treasures that the crusaders plundered from Hagia Sophia, so perhaps it was a Russian priest or monk, or maybe a Russian member of the Varangian Guard, who were expelled after the conquest. However, judging from the focus on the German involvement in Alexios’ scheme, the source may also have been German.

The Russian reaction seems to agree with certain Greek sources (like Nicetas Choniates) and Western European sources, that the arrogance of the Greeks was to blame. The Greeks probably could have defeated the crusade but they were more interested in petty infighting. Otherwise the Russians don't seem to have been very concerned at all; at least the city was still in Christian hands.

“It seemed almost a matter of indifference whether Constantinople was in the hands of the Greeks or the Franks. The Russians still remained aloof from the religious rivalry of Constantinople and Rome. The value of the Novgorod chronicle is its neutrality.” (Angold, pg. 11)

Sources:

The Chronicle of Novgorod, 1016-1471, trans. Robert Mitchell and Nevill Forbes (London, 1914)

Jared Gordon, “The Novgorod Account of the Fourth Crusade”, Byzantion 43 (1973)

Donald E. Queller and Thomas F. Madden, The Fourth Crusade: The Conquest of Constantinople, 2nd ed. (University of Pennsylvania Press, 1997)

Jonathan Phillips, The Fourth Crusade and the Sack of Constantinople (Pimlico, 2005)

Michael Angold, The Fourth Crusade: Event and Context (Routledge, 2014)