Greece managed to gain independence from the Ottoman Empire in the early 1800's because of the support it had from the West.
Greece was a territory that was of huge interest for the countries like Great Britain and France. They helped Greece with political propaganda, militaristic support, and financially in order to make it an independent country.
The Ottoman Empire was very weakened in this period, and its army was all over the empire because of the numerous rebellions, so Greece managed to gain independence relatively easy.
The first leader of the country was actually a German noble, which was later overthrown.