Well, this would be rather "easy" just have the Greek Civil War balloon out of control. Have the communists be more powerful, perhaps manage their resources better, extract more aid and help from more nations in the Warsaw Pact. At the same time, you need them to be not as strong that they actually take power. On the opposing side, there is the republican EDES, that has some fascist leanings and carries out ethnic cleansing in areas it controls, the National Army, supporting the Monarch but not the Cairo Government, as they see it as too liberal, and in come the Western Powers: the British support the Cairo Government, locked in a power struggle with the monarch, and they are mildly opposed to both EDES, and the monarchists, but very much opposed to the communists. The Americans are secretly arming EDES, and supporting them; fighting does not cease for a period of 5-6 years, and at some point the Americans fully send in the troops.
In the end, while it would seem plausible at some point that the communists could win, an uneasy coalition between the King, the Cairo Government and EDES sort of manages to control most of the territory, but there are still pockets of communist insurgents, republican radicals, etc.: there is no room for democracy or mellowing out the regime.
Alternatively, and if you'd like something closer to the current time period:
First of all, enhance police incompetence so that the terrorist organization "17N/17 Noemvri(17 of November)" is not destroyed in the early 2000s. Then, have the government of Costas Simitis be more committed to implementing sweeping changes to the Greek social security system, instead of effectively shying away; however, that leads to massive backlash, and protests sweep the nation. Some violent acts are carried out, and the aforementioned terrorist organization manages a high profile political killing or two. This massive backlash and an immediate danger the rule of law forces the government to completely change course in the end. So economic problems are not solved, but civil unrest is something that is there to stay.
The country then goes on to make a complete botch of the 2004 Athens Olympics, again with massive terrorist action. That worsens the political situation, and acts of political violence are starting to become more widespread. The election of Konstantinos Karamanlis does little to alleviate the problems, but they manage to win a second term in 2008, at whichpoint all hell breaks loose. Massive forest fires in many regions of Greece(IOTL they happened in 2007) press home the image of a govern,ent that is ineffective, unable to react to circumstances, and impotent. Then, Karamanlis tries to push forward educational reforms, and that begins the long descent into chaos: massive protest again break out, and all universities in the country are shut down by student protests. At some point, a 15 year old is killed by policemen(happened IOTL) and that lights the fuse. Athens becomes a battleground, and the government, instead of backing down, invokes Article 48 pg the constitution and suspends basic liberties, it brings the army into play.
But that's not enough, because as it is now 2008, ballooning debts and economic problems begin to rear their ugly head. An election is held, but there are massive irregularities. The opposition ekes out a win and a weak mandate, by trying to portray the former government as dictatorial nutcases, and that resonates. They run on a populist platform of lower taxes but massive ramping up of public spending. That clashes with reality, and by 2010, the government has to sign a much worse memorandum, with even worse terms. The prime minister, George Papandreou, feels he does not have the will of the people behind him, and stuns everyone by calling a referendum on the bailout; a couple of days before it, far right assasins kill the long standing leader of SYRIZA, an up and coming populist left wing party, Alexis Tsipras. His death makes him a martyr figure, and in this climate of huge political violence, the Greek people vote "no" to the bailout. Greece is kicked out of the EU and the Eurozone, and the government is reeling; the PM is removed by his own party, and the President of the Republic, having had to deal with a much worse and stressful situation, and being quite old, dies in a heart attack; elements of the army coalesce with the far right in order to gain power, and they do, but the whole country breaks down into civil war.