It’s doubtful the Greek population of Melbourne ever exceeded 300k; and the more Greeks assimilate, the harder that is to count. I think it could be argued that Chicago had a larger Greek population at least at one point. Thessalonica has a population over a million, so Melbourne was never the largest Greek city outside of Athens.
There was the occasional random Greek in Australia as far back as the 1820s. A discernible community in Melbourne dates from the 1890s (about 150 strong), and the first Greek church in Melbourne, the Church of the Annunciation, opened in 1901. Back then, the community was from a small number of Greek islands—Kalymnos, Ithaca, Kastellorizo, with one family member bringing another. Kalymnos and Kastellorizo were sponge diver islands, and diving was an initial source of employment for Greeks in Australia. (Hence the pearling houses of Paspaley and Kailis in Broome.)
The upsurge of the Greek population in Melbourne, and the rest of Australia for that matter, dates from 1950–1975, and was the result of mass migration of Greeks from impoverished post-war Greece (and in lesser numbers, Cyprus, Turkey and Egypt), to work in the factories and then small businesses of Australia. In my family, the earliest family member, my uncle George, came out to work as a carpenter in Sale in 1947; the latest family member was my uncle Andrew, who joined my father working in the fish ’n’ chip shop business in 1970.
Unlike the Italians, Greeks avoided becoming farmers in Australia, and stuck to the cities; for that reason, there was a healthy presence of Italians in rural Queensland, but the Greek population was heavily concentrated in the capital cities of each state. By far most Greeks ended up in Sydney and Melbourne; as of the 2011 census (Greek Australians), 30,000 people in NSW were born in Greece, and 50,000 in Victoria.
Melbourne was at the time the industrial powerhouse of Australia, and it needed factory fodder; that was one drawcard for early migrants, and I assume is what gave it the edge over Sydney. As with many other diasporas (and as indeed with the earlier Greek migration wave), family members brought other family members, and critical mass of the diaspora encouraged other Greeks to join it, as familiar territory. (The current Greek Financial Crisis wave of migration is conspicuous in Oakleigh, the current Melbourne Greektown, for both reasons.)