There is a two-fold reason for Russia's long-standing interest in controlling the city of Byzantium/Constantinople/Istanbul and the area surrounding it: one of symbolic value and one of practical value.

The beginning of something resembling a "Russian" identity came with the Christianization of the Viking princes of Kiev in 988. The princes, who had made their way from Scandinavia down to the modern area of Ukraine by way of Russia's rivers, ruled over the local slavic population and participated in a bit of "shopping" when they decided to look more seriously at the varied religions practiced by their neighbors. The Russian Primary Chronicle, our main and almost only source on this period, reports that the ban of alcohol and pork by Muslims living on the Volga river did not sit well with the princes. The dietary restrictions of Judaism practiced by the Khazars living in the Caucasus and modern southern Russia also did not appeal to the princes. Conversely, the envoys sent to Constantinople came back telling tales of the grand cathedrals of the city, famously reporting that "they did not know if they were in heaven or on earth." The princes of Kiev (and slowly but surely the slavic populations in their domains) converted to Orthodox Christianity and this point was used by later Russian princes in Moscow as a source of legitimacy to their rule. Especially after the fall of Constantinople in 1453, Russians took up the mantle of carrying on the traditions of Orthodox Christianity and in their minds linked themselves all the way to the Roman Empire through an inheritance described as passing from the Western Empire to the Eastern Empire and then on to the princes of Kiev and eventually the Russian princes of all the Russian city-states of the medieval period. Thus, Constantinople in the Russian consciousness formed a kind of source of civilization and legitimacy that made taking the city back from the non-Christian Turks a natural goal for Russian rulers. One could make a comparison to the western European desire to conquer the Holy Land/Palestine/Jerusalem because it was seen as the site of the foundation of their society.

In practical terms, Istanbul became an achievable goal with the expansion of Russia's territory into Novorossiya - modern southern Russia and eastern Ukraine under Catherine the Great in the 18th century. The new territory opened ports on the Black Sea to the Tsar and from there Russian goods could make their way into the Mediterranean and to profitable markets there. The only problem was the continued Turkish control of Istanbul and the straits that connected the Black Sea to the Mediterranean, the Dardanelles. Russia came to pursue an aggressive policy aimed at destabilizing Turkish power or winning directly through war control over the crucial straits and access to the Mediterranean. Saint Petersburg had only appeared in the time of Catherine's grandfather, Peter I, and it was Russia's first port and even it was unusable for nearly half the year because of the freezing of the Neva river. Without ports Russia could not participate in the type of commerce that was exploding in western Europe and strengthening the likes of England, France, and the German states. Those same European powers were aware of Russia's isolation and enforced it through the years as they warily eyed the massive land-locked empire on the periphery of Europe. In the 18th century, there was a kind of military parity between Russia and the Ottoman Empire and a number of border skirmishes and small wars did not see Russia ever threaten to take Istanbul, though they were slowly eroding Ottoman territory in eastern Europe and the Caucasus. In the 19th century, the waning of Ottoman power forced the hand of Britain and France to intervene and prevent the Russians from accomplishing their goal. Western European powers oversaw the creation of an independent Greek state, rather than a Greek revolt leading to a Russian invasion, and in 1853 Britain and France declared war on Russia to stop them from overwhelming the Ottomans and routing Russian forces in the Crimean War. Again in 1878, the western European powers, concerned by the settlement of the latest Russo-Turkish war with the Treaty of San Stefano forced Russia to renegotiate the settlement at the Congress of Berlin, once more preserving Ottoman territory where possible and creating independent buffer states to deny the Russian Empire access to the Mediterranean. To this day Istanbul still is in the sights of Russian geo-political ambitions, and the Russian search for ports that could be used year-round led to conflict with Japan in Manchuria in 1904-05 and can even be seen as a contributing factor to Russia's support of Al-Assad's Syria.

(Edit to not leave the last sentence hanging: The Syrians have been leasing a Mediterranean port to the Russian Navy in return for the Russian support and tech in the Syrian Civil War)