The current Prime Minister of Greenland is Aleqa Hammond. She is the leader of the Siumut Party, and became the country's first female prime minister after the 2013 elections.
Greenland gained the status of an autonomous Danish dependent territory with some self-governance and it's own parliament in 1979. One of Hammond's main policies is for Greenland to become a completely independent country in it's own right.
The Siumut party led by Hammond take a centre-left political view point, and have representatives in Danish parliament associated with the parliamentary group of the Social Democrats.
Greenland politics and their (fictional) Prime Minister feature in a couple of episodes of season 2 of the Danish political drama, Borgen, which gives you a real insight into the relationship between the two countries.
I would recommend this show regardless, however. It's great!