Only one Greek King, Otto I, was German. His successor, George I, was Danish, and all of the subsequent Kings were descended from George.

Anyway, as to why, two reasons. Firstly, there wasn't really an indigenous Greek nobility - there were people who were richer and controlled more land than others, but they didn't really constitute an aristocracy in the sense that it's usually meant. The closest thing to a Greek monarchy were the clan leaders, but given the rivalries of the clans, they would have made for poor monarchs, since chosing a monarch from one clan would have created antagonism from their traditional rivals.

Secondly, and more importantly, the Greek people didn't actually choose their own King. The end of the Greek war of independence came from the decision of the three concerned great powers - Russia, France, and the UK. It was they who pressured the Ottomans into ceding Greek independence, and they were able to impose a Bavarian (technically not German, since Germany didn't exist as a state at the time) monarch on the Greek people.

Interestingly, when the Greeks first chose a head of state, they did choose a Greek person - Ioannis Kapodistris, a former Russian foreign minister. He became quite unpopular due to his feuding with Greek clans, so when the Great Powers informed the Greeks that they had chosen a German ruler for Greece, the national assembly complied.

Otto subsequently became very unpopular and fled Greece, but when he was replaced, there was little call for a native Greek head of state, so another foreign candidate was found.

This was a somewhat common practice during the era between the Napoleonic wars and World War I - other newly independent countries would often import a monarch from overseas. The Belgians chose a German ruler (also a former Russian civil servant) as their King, the Albanians had a German Prince picked out for them by the Great Powers (although he also didn't last), and the Finns planned to import a German ruler in 1918 until events overtook them.

Source: Potatoes Of Kapodistrias: Modern Greek History by Gregory Zorzos