The Ottoman empire had been crumbling for at least a century before it finally disintegrated in 1918. For centuries, Russia, "the Third Rome" had desired to gain posession of the city. In 1915, the British and French agreed to let the Russians take posession. The Bolshevik Revolution of 1917 changed that.
The Greek state, having expanded greatly since gaining independance was looking to return to it's old capital. In Greece,this was referred to as the "Megali Idea", the dream of reclaiming the Byzantine capital. Greece's entry in WW1 on what turned out to be the winning side made this a realistic ambition.
Greek Prime Minister Eleutherios Venizelos based Greece's claim to Constantinople at the Paris peace conferance in 1919 on two principles. 1) American President Woodrow Wilson's fourteen points which stressed self determination for nations in the Ottoman empire. 2) Venizelos pointed out in a memorandum that the "principal native element of the population of Constantinople is Greek, being numerically greater than all the other nationalities put together except the Turks." Quote taken from a New York Times article of January 16, 1919.
In the end Greece liberated Smyrna and the western portions of Asia minor as well as Eastern Thrace. The Greeks of Pontus sought to form a state of their own. Greece was not allowed to enter Constantinople, although the idea of Greek claims was considered and ultimately rejected. In September 1920,Venizelos petitioned the allies to turn Constantinople over to the Greek army, and in return the allies refused (From New York Times article of September 20, 1920.)
In July 1921, Prime Minister Gounaris argued that Greece had the right to march on Constantinople, which the allies had by then intended to preserve as a Turkish capital, as a result of Greek military victories up to that time. The allies countered that their respective armies numbered 10,000 soldiers and all would be used to resist a Greek march on Constantinople. (From New York Times article "Talk of Greek March on Constantinople". July 27, 1921.)
The Greeks continued to hope that they would enter Constantinople up until Kemal Ataturk's armies entered in the fall of 1922. It is interesting that when the British entered Constantinople following the Ottoman empire's collapse Greek hopes were high. Greek flags rose in Constantinople. The The Patriarch Meletios IV lobbied the allies to support union with Greece. It is ironic that the Greeks of Constantinople and Asia minor survived into the twentieth century only to disappear almost entirely by the twentyfirst when the world had become more enlightened to the plights of minorities and human rights.