Justinian had tried to force an end to post-Chalcedon problems arising from the Oriental Orthodox (who accept the first three Councils but not Chalcedon) by condemning a body of works that seemed to fit the Oriental Orthodox view, which became known as the Three Chapters. (There are more than three works in the Three Chapters, but it gets its name from the fact that there were three authors involved -- Theodore of Mopsuestia, Theodoret of Cyr, and Ibas of Edessa. All three were already dead.) He began to get the bishops to agree to it, by more than a little pressure and occasionally by a bit of roughing up. Many of the Latin bishops, even those who were in the East at the time, refused to do so -- some out of sympathy for the works, but most for the eminently reasonable reason that they didn't read Greek and so had no idea what the books actually said. What they did know was that all three authors were highly respected by other highly respected authors.
The Council did not start on a good footing at all, at all. Pope Vigilius was on his way to Constantinople, in part because Rome was not a particularly safe place at the moment; his journey went through precisely the territory that was most vehemently opposed to the condemnation, so he must have heard about it in negative terms from virtually every bishop on the way. At Constantinople, he started excommunicating people who had signed off on the condemnation. Then he issued a judgment condemning the Three Chapters himself. (It's usually thought that someone had given him actual translations.) Then he withdrew it. Then he met with Justinian and agreed that they would resolve it with a council. Justinian then went and issued another condemnation on his own authority, which led Vigilius to issue an encyclical complaining about Justinian's behavior. Another agreement to have the matter decided by a council was reached. Then Vigilius decided against it and issued a letter to everyone telling them that he would not recognize any council that proceeded without him.
The Council proceeded without him. The Three Chapters were officially condemned. And in the Third Session, the Council ordered Pope Vigilius's name struck from the diptychs -- effectively this means that it ordered the liturgy to proceed as if he were no longer bishop of Rome. (Sometimes it's treated as a 'deposition' or an 'excommunication'; these terms are both too strong, as is seen by the suspiciously common excuse in much later days that a pope's name was left off the diptychs entirely by mistake. If someone can get out of it by saying, "Oh, wow, how did that happen?" it's an exaggeration to treat it as being in itself an excommunication.) The emperor then had him imprisoned. After about six months in prison, Vigilius reversed himself again, claiming that he had been misled by his advisors, and condemned the Three Chapters again -- but very notably he did it on his own authority and without any mention of the Council. Unfortunately, a lot of Latin bishops were still in the same position they were. Very important Western sees like Milan (arguably the second most important see in the West) broke communion with him over it. News of the council did not travel very widely in the West, which is why for a very significant part of the early Middle Ages the Spanish numbering of the councils skips over Second Constantinople.
In the East, the Council failed utterly to do what it was supposed to do, namely, eliminate the schism that had developed over Chalcedon. Not only were the Oriental Orthodox not inclined to pay attention to it if they did not have to do so, the entire East was beginning to be inflamed with war, first from Persia and then from the sudden rise of Islam, and everyone in the far east of the Empire had more immediate things on their mind.
It was not a shining example of an ecumenical council; it was in many ways exactly what you don't want in a Church council. It was not a shining example of papal wisdom and fortitude; Vigilius was certainly no Leo. It was not a shining example of Imperial defense of the faith; Justinian managed to flop massively at this point. Emperors and popes, saints and bishops, managed to do almost everything in the most stupid way available. But the Council did become a major element of Byzantine theology. And acceptance of it as an ecumenical council slowly spread over the next several centuries, at least in the West. The Second Council of Constantinople has the distinction of being perhaps the least immediately effective ecumenical council ever. But when it became secure, it became very secure.
From the Sentences against the Three Chapters issued by the Council:
Our great God and saviour Jesus Christ, as we are told in the parable in the gospel, gives talents to each one according to his ability, and at the proper time asks for an account of what has been done by each one. If the person to whom only one talent has been given is condemned because he has not worked and increased it, but has only preserved it without diminishment, how much more serious and more frightening must be the condemnation to which the person is subjected who not only fails to look after himself but scandalizes others and is a cause of offence to them? It is clear to all believers that when a problem about the faith comes up it is not only the heretical person who is condemned but also the person who is in a position to correct the heresy of others and fails to do so. To those of us to whom the task has been given of governing the church of the Lord, there comes a fear of the condemnation which threatens those who neglect to do the Lord's work. We hurry to take care of the good seed of faith protecting it from the weeds of heresy which have been planted by the enemy.
It's perhaps worth noting that Second Constantinople occasionally pops up in unusual places as people rediscover it. It has been argued for instance, that Thomas Aquinas shifted his account of prophecy in the Psalms (one of the things that comes up incidentally in the discussion of Theodore of Mopsuestia) after having come across a better translation of the condemnation. When Nikodemos the Hagiorite insists that Augustine of Hippo is a saint, it's on the basis of the fact that the Second Council of Constantinople treats him as such -- and indeed, among those Eastern Orthodox who have since come to recognize Augustine as a saint, it is a combination of the authority of Second Constantinople and the Hagiorite himself. And it occasionally shows up again in theological discussions of synodality and episcopal collegiality. A most curious council in both its manner of proceed and its effects.