The Russians have always coveted Istanbul/Constantinople because it is the gateway to the Mediterranean. With the Bosporus and the Dardanelles closed as they were in the First World War with mines and submarines, the Russian Black Sea Fleet was stuck.

Furthermore, the Russians relied heavily on the straights being open for trade, with a large percentage of their exports passing through the straits.

Finally, Imperial Russia considered itself the successor state to the Byzantine Empire through the Russian Orthodox church, thereby solidifying their claims to Constantinople and the straits.

So, with these claims, why did Russia ultimately get promised Constantinople and not Greece? Russia was a Great Power that had imperial ambitions concerning the city and the straits. Once the Entente powers won, the Central Powers would be carved up, especially the "sick man of Europe," the Ottoman Empire. Russia gets Constantinople, France gets Syria (modern day Syria, Lebanon, and Jordan), and Great Britain gets Mesopotamia, Palestine, and gets confirmed in their occupation of Egypt.

All of this took place except Russia had themselves a little revolution and negotiated the Treaty of Brest-Litovsk, losing a bunch of land before it being reinstated in the Treaty of Versailles.

TL;DR- Russia wants security for trade, Black Sea Fleet gets access to Mediterranean, plus they get their prize for being a good partner in the war.

The short answer is that the Greeks were neutral at the time. Why would Britain, France and Russia agree to give the capital of the Ottoman empire and one of the most strategically important cities in the world to a neutral power?