From Roger Crowley's 1453, I can say that it was important to Muslims in general that they capture Constantinople, and it was in fact prophesised for Muslims that Constantinople would be captured. I have the ebook, so I'll copy sections out:
Byzantium had proved the most obdurate of enemies, and Constantinople itself remained for Muslims both a scar and a source of deep longing. Many martyrs had perished at its walls, including the Prophet’s standard-bearer Ayyub in 669. Their deaths designated the city as a holy place for Islam and imparted a messianic significance to the project of its capture. The sieges left a rich legacy of myth and folklore that was handed down the centuries. It included among the Hadith, the body of sayings attributed to Muhammad, prophecies that foretold a cycle of defeat, death, and final victory for the warriors of the Faith: “In the jihad against Constantinople, one third of Muslims will allow themselves to be defeated, which Allah cannot forgive; one third will be killed in battle, making them wondrous martyrs; and one third will be victorious.”
As far as Mehmet II himself, he was somewhat religious but not fanatically so. My understanding was that he wished to put an end to Constantinople more out of fear of the European powers attacking the Ottoman Empire in its defence, and also because the Ottomans had failed before to capture it and he was exceptionally ambitious. The term used often in the book is "the red apple", a tantalising goal for Ottoman rulers. It's also called a "bone in the throat of Allah", because the Ottomans controlled the territory around Constantinople but could not capture the city itself. Jihad was a factor. Here's another quote describing Mehmet II in his youth:
“It is clear that even at an early age Mehmet was magnetically attracted to the project of capturing the Christian city and making himself heir to the Roman Empire. In a poem he wrote that “my earnest desire is to crush the Infidels,” yet Mehmet’s longing for the city was as much imperial as it was religious, and derived in part from a source that was surprisingly non-Islamic. He was deeply interested in the exploits of Alexander the Great and Julius Caesar. Alexander had been transformed into an Islamic hero by medieval Persian and Turkish epics. Mehmet would have known of Alexander from his early years; he had the Greek biography of the World Conqueror by the Roman writer Arrian read to him daily in the palace. From these influences he conceived for himself twin identities – as the Muslim Alexander whose conquests would reach to the ends of the earth, and as a gazi warrior leading jihad against the infidel. He would reverse the flow of world history: Alexander swept east; he in his turn would bring glory to the East and to Islam by conquering the West. It was a heady vision, fueled by his personal advisers[…]”
The introduction of gunpowder also made it much more feasible to breach Constantinople's famous and formidable walls, and these were used heavily in the city's ultimate downfall. Further, most Christians did not care about Constantinople by this time, particularly because of the schism between Orthodox and Catholics, and Constantinople had become very poor and unpopulated, and largely unable to protect itself from the overwhelming force the Ottomans could muster.
In spite of that, it performed remarkably well in holding off the Ottoman conquest for one month, three weeks and two days. It's quite a riveting read.
There was a religious aspect to be seen, as Mehmet II prayed in the Hagia Sophia straight after he entered the city and very quickly set about converting it into a mosque, but overall, there was less fervour than there could have been in the ruler's conduct. Mehmet II's son Bayezit II was much more religious and reportedly said, much later down the line, that his father did not believe in Mohammed.
I hope it's acceptable to use these long extracts.