It seems like in the event of war, the entire fleet would be occupied in trying to take the Bosphorus, which I imagine would have been nearly impossible considering the location of Istanbul. It just seems like the entire fleet would be constrained to just the Black Sea.
I guess what I'm trying to say is it seems like it would be a very long time before this fleet could be brought to bear in the Mediterranean or Atlantic and by then these areas would be strongly secured by members of NATO. So what was the purpose of the fleet? purely defensive?
The Black Sea Fleet contained a number of landing ships. It was trained in supporting amphibious operations including paratrooper and beach assaults.
The purpose of the fleet was to threaten NATO's member state Turkey and to support Black Sea allies like Romania and Bulgaria. It was also trained to operate in a post-nuclear environment, when naval operations would be the easiest way to move supplies and troops between points with devastated infrastructure.
Since the Dardanelles were open to the Soviets by international law during peacetime, the Black Sea fleet was also able to visit Soviet allies in Africa and the Middle East. Massive shipments of military and economic aid were launched from the Black Sea's ports. Russia still maintains a base at Tartus, Syria and the Black Sea fleet is vital to that base's operations.
Edit: /u/tenthreeleader does a good job in their reply of covering the strategic portion of the answer, as opposed to my narrowly focused response.
Because without the ability to move into and out of the Black Sea, Russia has no easily accessible 12-month warm water port. When you add in that Russia and Turkey (and their predecessors) have been in some sort of conflict or rivalry since 1568, you see why the Soviets felt it necessary to have a strong fleet in the Black Sea.
Globalsecurity.org notes that the Russian Arctic ports are icelocked much of the year, Kamchatka has a four-month shipping season and Vladivostok is icelocked for four months out of the year. Europe blocks their access to the west, and Port Arthur is 4,400 miles from Moscow by rail.