The best place IMO - is the Napoleonic Wars. Let Napoleon storm in, but in 1812, rather than go into Russia - instead see if he can propose a treaty - perhaps a Russian secession of territory to Napoleon, or an alliance, to invade the Ottoman Empire, where neither the Russians or French would be granted the crown but a joint invasion to provide territory for a new Roman Empire - centred around the Greek independence movement. Whoever would be chosen as King, is instead offered the title of Roman Emperor, with the captured territory consisting of the Greek Territories (See Greece/Constantinople, S.Balkans), E.Balkans north of the Rhodopes 'Administered' by the Russians on the Emperors behalf, and the W.Balkans 'Administered' by the French.
Whatever butterflies this has on Europe is for Europe, but lets assume that post-war (say 2-5 years?), Russia and Napoleon still go to war, leading to conflict between the French and Russian Administrations - who are told by the new Emperor (backed by a rapidly expanding and modernising Roman Army) that if they leave, and do not fight on Roman territory, that the Roman Empire will stay neutral in the war. Seen as advantageous by the French and Russian forces, who rapidly withdraw to new frontlines, the Roman Empire starts to establish its Orthodox, Greek Bureaucracy, reinstating the old Roman system of recruitment into the army being the main path to citizenship, with the army operating using Greek.
With northern Neutrality confirmed, and the rise of the new Roman Army, the Romans can try to turn their attentions East - with an Ottoman Empire not only trounced, but wracked by nationalist rebellions, the Romans can invade and try to take territories populated by Greeks. With the army-based route to citizenship, paid for using land that was abandoned by Balkan muslims moving east.
It will have problems, most certainly, how to ensure that the other cultures, whilst being Hellenised through the army (and only the army), don't rebel.
Now, admittedly there isn't a Wallachia or Bessarabia in this Empire - and I doubt it would be ready to take that territory from Russia any time soon - but through some sort of federalism, and democracy - I'm imagining a Constitutional Monarchy, over a Thematic Democracy, or a Democracy of the Citizenry (which is attained by military service), which would mean the entire legitimate demos would speak enough Greek to be in the army - but is still allowed to use their native tongue at home. Combine it with land-payments that could scatter demographics around enough that speaking Greek is more convenient, as per the Romans of old, and you can rapidly expand the state until it has to rely on paying troops using salaries rather than land payments.
The Emperor may well step down, or become increasingly powerless as per the British Monarchy, serving as a tourism attraction and source of national unity - neither really matters.
The only concern I still have is whether the Romans can support a Federal Multi-cultural Empire, whilst at the same time creating Greek as the Common Language. I don't see it as impossible, but it won't be easy.