Specifically, I was wondering about that since traditionally Crimea had been the zone of Greek and Byzantine economic interests and even control, and the Genoese were obviously their big competitors; however, for the Italians to get there, they must have passed the Bosporus, and not just once: with all the trade volume between the two province of the Genoese Republic, there must have been Italian ships passing Constantinople back and forth all the time...why did the Greeks allow that?
Also, just how firm with their grip of the peninsula (as in a simple trading outpost, or more of a colony)? What kind of challenges did they face and how beneficial was the whole affair for them?
The Byzantine empire was conquered by the fourth crusade in 1204. The emperor fled to the Asian mainland and tried to reconquer the capital from the new seat of Nikaia. Since the fourth crusade was mainly supported by Venice, he allied himself with Genua who was to support the reconquest with naval forces. After the reconquest - at which the Genoese navy wasn't used - the empire granted trading rights to Genoese merchants. Genoa used these rights to dominate trade in the eastern half of the Mediterranean. They ventured into the Black Sea and bought the city of Caffa from the Golden Horde.
Some historians hold the view, that the Black Plague came to Western Europe via Caffa in 1347. But in general, the colony was the main trading hub between Catholic Europe and the lands around the Black Sea.