The Russian goal was always Constantinople. The Treaty of Berlin in 1878 put a dent in their aims after almost bankrupting themselves during the Russo-Turkish War (1877-78). The one bright star to come out of it was that the Tsar got a free hand in Bulgaria. He hoped this would be the ally that would provide the leverage in the Balkans. The Tsar chose Alexander of Battenberg, and hoped he would be a suitable puppet, but when Alexander started making policies based on the best course for the Bulgarians, rather than Russia, he was kidnapped by Russian agents in 1887 and forced to abdicate. The subsequent Prince Ferdinand, chosen by the Bulgarians, proved no more palatable to the Tsar. There were thus further intrigues to remove the new prince so that the Tsar could once again try to install a regime favourable to Russia. Russia employed a disaffected Bulgarian officer, Major Panitza, to assassinate Prince Ferdinand and his prime minister, but the plot was foiled in February of 1890 and Panitza was executed.
Another shock for Russia was towards the end of the war with Turkey when Britain sent their Mediterranean fleet into the Sea of Marmora to deter further Russian advances. The Russian Black Sea fleet was almost non-existent at this time and could have done nothing to prevent the British from sailing into the Black Sea and wreaking havoc along the Russian coast. From this time on the Russians were determined to build a powerful Black Sea fleet, but they soon realized that even that would be unable to stop the Royal Navy from entering the Black sea at will. Following a further war scare with Britain in 1885, the Russians realized that in order to safeguard the Black Sea, they would need to take control of the narrow Bosporus straits at Constantinople. They made plans to capture the straits in a ‘coup de main’, by landing troops on either side of the seaway, supported by the guns of the Black Sea fleet. These soldiers would capture the Turkish forts, and install naval artillery to protect minefields to be laid in the straits. If the Royal Navy attempted to force the straits, then whatever ships survived the mines and naval artillery, could then be dealt with by the Black Sea Fleet.
The interesting point is would a truly pro-Russian Prince, either in 1879, 1887 or 1890 have led the Russian conquest of Constantinople? The lapse of the reinsurance treaty with Germany in 1890, against the wishes of the Tsar, ended the assurance of German neutrality in the adventure. The Tsar would therefore need the Bulgarians to engineer an incident with Turkey before that occurred. I believe that the Tsar wanted to use the annexation of Eastern Rumelia as the trigger, allowing Russia to step in to support Bulgaria. Unfortunately, the hated Alexander pulled that trigger, and the Tsar ended up being the one to protest that this broke the terms of the Berlin Treaty! It would also need to occur when the Black sea fleet was strong enough to support the operation, which didn't occur until 1890. Someone above mentioned it being a small window of opportunity, it would seem to be a small window indeed, and only if Bulgaria is pro-Russian.