Historians don't have much to judge concerning Constantine XI's reign as Emperor, with the main reasons being his short 4 year reign and the sheer hopelessness of his situation he was in. By the time he ascended to the throne, there was really nothing that even the most energetic Emperor could do to stop Mehmed II's imminent attempt to conquer Constantinople.

Constantine XI did show competent leadership potential during his post as Despot of the Morea, the Byzantine's Empire holdings in the extreme south of Greece, where Constantine XI invaded the small Duchy of Athens and forced Nerio II Acciaioli, the Duke of Athens, to pay tribute to him, where he had been paying tribute to the Ottoman Sultan previously. Ottoman Sultan, Murad II responded to this by raiding the Morea with a punitive force of 60,000 men and overran Constantine's defenses, but Athens remained in Constantine's control by the end of the raid. Constantine's victory over Athens would be the last victory the Byzantines would ever see.

By the time Constantine became Emperor however, there were two main issues concerning the fate of Constantinople; Mehmed II's impending attack on the city upon his ascension as Sultan, and the ongoing negotiations for a reunion between the Catholic and Orthodox Churches. If this rift between the churches was successfully healed, then the Byzantine Empire would find protection from the Ottomans by the Pope and the other Catholic Kingdoms, but this involved recognition of the Pope as the primary authority of the church, which was staunchly rejected by Orthodox Clergy.

Constantine XI himself was raised as a Catholic and was a strong supporter of the reunion of the churches along with Gregory III, the Patriarch of Constantinople. The Council of Ferrara-Florence in 1439 was held under the previous Emperor, John VIII, where the Emperor and Patriarch agreed to accept the Pope's authority along with the doctrines of the Filioque and Purgatory. This agreement, however, was rejected by Orthodox Bishops and the Empire's civil authorities, destroying hopes of a union between the churches in time. Constantine XI tried to revive this agreement, but to no avail as it continued to be rejected by the Empire's civil administration and the Bishops.

By 1451, all that Constantine could do was keep the city's defenses in working order, stockpile food, water, and weapons to keep the city supplied and armed, and call out to Catholic Powers for aid against the coming siege as Mehmed II erected fortifications and raised an army of 80-100,000 for the siege.

With only token help coming in from the Republic of Genoa after Constantine's pleas for aid in the coming siege, there was really nothing that could be done to save the Empire, after a nearly 2 month siege, Ottoman forces broke through the gates and conquered the city.

After the Fall of Constantinople, Constantine XI became a figure of legend in Greek Culture, with stories that he removed his imperial garments and charged into the gates where he met his demise against the Ottoman army. There's really no solid source telling us that this happened but people have just accepted it, as Constantine's body was never recovered or clearly identified after the battle. The Ottoman's chopped off the head of who soldiers believed to be the late Emperor along with dragging the corpse through the streets of the city, but nothing confirms that he was the Emperor either.

As for sources, I'd definitely check out Foreign Relations and the End of Byzantium by Adam William Hellebuyck as well as The Immortal Emperor: The Life and Legend of Constantine Palaiologos, Last Emperor of the Romans by Donald M. Nicol