After Russia's loss in the Crimean war in 1856 Russia was forbidden to have any warships in the Black Sea by Treaty of Paris, leaving one of the most important parts of Russia - its only non-freezing sea ports and the area through which most of Russia's sea trade went, unprotected. Hence in the late-1860s Russia started to think about abrogating the terms of the treaty in order to protect its coast against rapidly deteriorating relations with Turkey. It was obvious that in the beginning the strategy for the Russian fleet would be purely defensive against relatively powerful Turkish fleet and so in the 1869 it was decided to build armored warships to protect the coast and the ports. This where the Rear-Admiral Popov comes in, in the 1860s he was the unofficial head of the shipbuilding in Russia.
It was during the design discussions, Popov suggested an idea that he had for a while - building round ships which would have carried heavy armor and armament on smallest draft and displacement. To prove the concept a small round steam launch with a diameter 3.35m was built. It proved successful enough for the construction of the full size ships to proceed. In 1871 with the defeat of France in the Franco-Prussian was Russia denounced the Treaty or paris and set about restoring it's Black Sea fleet.
Initially 6 round ships were planned, but to begin with two were laid down in St. Petersburg in the beginning of 1871. If you are thinking that this sounds wrong as St. Petersburg is on the wrong sea, you are correct, but since the south of Russia was really underdeveloped in those years the ship hulls had to be laid down and built in St. Petersburg, then disassembled, shipped to the Black Sea port of Nikolaev in pieces, and then assembled again. This caused the immeasurable delays as the railroad to Nikolaev was only completed in the middle of the construction and sometimes reached absurdity, when it proved easier and cheaper to buy Russian wood in England and have it delivered to Nikolaev, rather than to ship it from the interior of the country. Nonetheless the construction of the first ship, called Novgorod, started on 17 December 1871, while the second one, Kiev, was laid down in January 1872. Novgorod's construction proceeded relatively smoothly but Kiev became a platform for Popov's experiments - new, lighter engines were ordered for it, armament and armor was increased and in the end the construction was stopped to await testing results of the first ship. Due to a bad financial situation in Russia at the time the other four ships were at first postponed and then cancelled all together.