First of all, as a PoD, I think the easiest way to do it is by killing/removing and exiling King George II sometime between 1936 and 1939, plus Metaxas (the dictator-Prime Minister) somehow get convinced that Greece's interests lied with the Axis instead of the Allies. Thus, you butterfly away the British guarantee.
It's actually very easy to keep Greece out of the Allies, but tipping it to the Axis instead of neutrality is quite a challenge. The country's leadership would want to be promised one or more of Northern Epirus (from Italy-held Albania? No.), the Dodecanese (from Italy? Again, no.), Cyprus (Nobody sane would think the Axis can deliver that.).
Perhaps Mussolini teams up with the new Greek dictator (after both George's and Metaxas' deaths) to split Albania before the Italian invasion, Greece getting Northern Epirus. Greece would probably ask for nothing more.
Assuming it goes with the Axis, many of the following these may happen:
-The Germans actually do send the paratroopers to take Malta (although I wouldn't bet the house on their success).
-Yugoslavia is not couped and remains Axis-friendly.
-Bulgaria probably stays neutral, at least for a longer period of time.
-Greece would send token forces to Barbarossa, but no more than 1-2 divisions.
-I don't know about the actual effects on Barbarossa itself, but Germany did lose machinery, oil and time in the Balkans.
-Both Italy and the Commonwealth are stronger in Northern Africa, but the balance will tilt heavier towards the Allies, maybe they reach Tripoli by fall '41.
-By the end of the war, the Soviets invade the northern part of the country and the Allies land on Athens, the south and the islands. As said above, that could lead either to a West-East Germany scenario, or the creation of a neutral Greece (unlikely).