Now, I am not an expert in the exact nature of your question - particularly the question of what these two empires did in efforts to substantiate their claims as "The Third Rome", but I DO know a lot about the history of the Ottoman Empire and the Russian Empire and their involvement with one another. In regards to the Ottomans my areas of expertise are their methods of warfare and the politics involved therein as well as the nature of the Sultans, (specifically Murad I, Murad II, Mehmed II, Selim I, and Suleiman I - the Sultans that oversaw the largest territorial expansions by the Ottoman Empire). The Russian Empire on the other hand I believe I am a bit more familiar with, though if there is one aspect in regards to your question that I am the most familiar with it would be the Fall of Constantinople itself.

Firstly, the Ottomans never truly called themselves the "Sultanate of Rum (Kingdom of Rome)", which was actually a precursor to the rise of the House of Osman. The Sultanate of Rum would peak in the 1200's and then begin to disintegrate until the early 14th century, paving the way for the rise of the House of Osman and other powerful Beys of the time, who ruled their own distinct post-Seljuk territories like the Beylik of Aydin and the Beylik of Karaman. Despite the beliefs of many, the rise of the Ottomans was far from guaranteed. While it is somewhat clear that the Anatolian Turks were on the rise, and the Greeks to their west were on the decline, there was no obvious path to follow when it came to unifying the disparate Turkish Beys. The House of Osman would grow from a small border state in western Anatolia (modern-day Turkey) into a formidable nation-state by consolidating neighboring Beys into a single political entity. They would expand westward into Europe through the use of Ghazis, which they used as a social, political, and military tool that enabled them to promote holy war as a means to an end.

A large factor in the early rise of the Ottomans was their narrative of Holy War against the decadent Westerners, which led to the Fall of Constantinople and, eventually, advances as far as Budapest, Hungary and the gates of Vienna, Austria. Mehmed II "the Conqueror" and other Ottoman leaders were pioneers in the use of many differerent aspects of warfare - especially the use of cannon. They maintained a flexible, diverse army that consisted of different corps or divisions with various functions like raiding, scouting, flanking, archery, shock assault, siege warfare, etc., and all of these branches were supported by a well-structured and elaborate supply network. As a result, the Ottomans arguably had the most modern army on the planet from around 1450-1600, which is when we begin to see incredibly disciplined, professional armies arise in other parts of Europe as well. Not bad for a group of warriors who started out as what was basically a March (a vassal buffer-state) in western Anatolia!

In regards to your question, the only aspect of considering itself as a successor to the Roman Empire that I am aware of in detail is Mehmed II's claims of being the "New Rome." It is well-documented that his motivation for conquering Constantinople (now Istanbul, which is simply the Turkish way of saying "to the city" in Greek!) was that he wanted the prestige of the Eastern Roman Empire to be attached to his dynasty and his state. He did not want to simply control Byzantine territory - he wanted people to definitively identify his country as a continuation of the earlier Roman Empire. Their is some confusion nowadays due to terms like the "Byzantine Empire" that leads some to believe that there is a very sharp contrast between the earlier Romans and the Eastern Roman Empire that finally collapsed in 1453, but this has only ever been true in the eyes of the Latin West, who recognized the Holy Roman Empire in Germany as the successors to the the Roman Empire. The inhabitants of the Balkans, Greece, North Africa, the Middle East, Central Asia, and Eastern Europe more accurately recognized the "Byzantines" as the same polity as the earlier Romans.

As an interesting note, during the unification of Greece just decades ago there were people on some islands throughout the Aegean that, when asked if they were Hellenes (the Greek name for Greeks), replied with "No, we are Romanoi (Romans)." In essence, pretty much everyone who knew about Constantinople apart from the Latin West considered it to be the capitol of the Roman Empire. Thus, when the Turks finally took the city in 1453 it was obvious that they had supplanted the Greeks as the successors, albeit of a different religion and with an ethnically foreign ruling class, of the Roman Empire of yore. Mehmed II took the title of "Caesar of Rome" after his successes in 1453 in order to further cement this idea and he gave his new Greek citizens many rights and privileges, particularly freedom to continue practicing their faith much the same as before. He wanted them to consider themselves citizens of his empire, not a conquered people.

Which leads me to the Russian Empire. Beginning in the mid 10th century with the conversion of the Russian lands to Christianity by Greek Orthodox monks, Russian religious leaders began to espouse the idea of being part of a greater Orthodox Christendom with its heart in Constantinople. For centuries, Russian cities would request patriarchs and priests from the Eastern Romans in an effort to maintain the "true faith." When Constantinople fell, Russians were shocked. Though the Eastern Orthodox Church would carry on much the same as before, Russians saw the collapse of the Eastern Roman Empire as a foreboding portent. In their eyes, someone had to take up the mantle of true Orthodox Christianity. Before Constantinople fell, the Byzantine Emperor had signed an Act of Union that placed his church under the authority of the Papacy in an effort to garner military assistance from the West, which was a wasted effort. Most Orthodox Christians were disgusted by this and felt betrayed, with some even claiming that they would rather see their Church destroyed than placed under the guidance of heretics. Many others saw the Ottomans as potential protectors who would allow their Church to survive unabated and unchanged, which turned out to essentially be true. To the Russians, though, Orthodox Christianity was in dire straits and some polity had to assume the role of being a guardian and mother to the Church. A few decades after Constantinople fell to the Turks, in a letter written by the Russian monk Philotheus, it was asserted "Two Romes have fallen, the Third stands, and there shall not be a Fourth." Thus, if Russia were to ever be conquered by foreigners or lose its faith for some reason, then that would truly signify the end of the original Roman Empire. Religion played a very central role in people's ideas of what constituted a "people" or a "civilization" throughout most of European history.

Basically, the heart of the answer depends on an understanding that "Rome" was an interchangeable term depending on who was using it and that there was a fundamental understanding that "Rome" meant the center of not just a nation-state, but a people and a faith. For the Ottomans, it signified the center of the greatest civilization Europe had seen, whereas in Russia it signified the center of their pure, Orthodox faith.

Sources:

The Fall of Constantinople by Steven Runciman

The Ottoman Empire: 1300-1600 by Halil Inalcik

Destiny Disrupted: A History of the World Through Islamic Eyes by Tamim Ansary

The Life of Archpriest Avvakum

A Concise History of Russia by Paul Bushkovitch

Natasha's Dance: A Cultural History of Russia by Orlando Figes

Peter the Great: His Life and World by Robert K. Massie

Catherine the Great: Portrait of a Woman by Robert K. Massie

Follow up question. The turks controlled Constantinople, which of course was extraordinarily important in terms of the Roman sphere of influence. But what claim did the Russians even have to this? Was it orthodoxy? If it was why do we not see similar claims from regions like great Moravia?