When Greece regained independence in the 1830s, there was the need to find a candidate to become king. Otto, the second son of the King of Bavaria, was selected. However he proved to be unpopular and was deposed in 1862.

As the Greeks wanted closer ties to the Great Powers, they overwhelmingly supported Prince Alfred, the son of Queen Victoria of the UK. However this was forbidden by the convention that established Greece, as no Great Power was to have a relation on the Greek throne. Thus through some negotiation they settled on Prince William of Denmark, the second son of Prince Christian, the heir to the Danish throne. William took up the slightly modified title of "King of the Hellenes" (Otto was simply "King of Greece") and the regnal name George.

It is also worth noting that this was the first such case of a Balkan kingdom being founded and led by a European prince. When both Romania and Bulgaria broke away from the Ottoman Empire, they each took a German prince as their king (Carol and Ferdinand, respectively; notably the former was a Hohenzollern, related to the German emperor, while the latter was a Saxe-Coburg-Gotha, which was related to both the British and Belgian monarchs).

As for the second part of the question, I believe you are referring to the 1924-35 republic. That was established following the Greco-Turkish War, which ended poorly for the Greeks and led to the establishment of modern Turkey. The defeat was blamed on the monarchy, who were unpopular already due to their conduct of the First World War, and were consequently overthrown. However the republic didn't fair any better, and combined with the Great Depression led to the re-establishment of the monarchy in 1935 after a military-led coup.