Hamelin was born in St. Vital, Manitoba (now part of Winnipeg), and was educated in St. Boniface and at the Juniorate of the Sacred Heart in Ottawa. He then returned to Manitoba and worked as a merchant, becoming manager of the Joseph Hamelin Co. general store in 1912. He served as reeve of Ste. Rose from 1900 to 1914.
Hamelin was first elected to the Manitoba legislature in the 1914 provincial election. Running for the Conservative Party in Ste. Rose, he defeated Liberal candidate J.A. Campbell by sixty votes. After the election, he served as a backbench supporter of Rodmond P. Roblin's government.
Hamelin's time on the government benches was brief. The Roblin government was forced to resign amid scandal in early 1915, and was resoundingly defeated in the 1915 provincial election. Hamelin was one of only five Conservatives to win re-election, defeating Liberal candidate Z.H. Rheaume by twenty-nine votes. He was again returned in the 1920 election, in which the Conservatives won eight seats out of fifty-five.
In the 1922 election, Hamelin won re-election over United Farmers of Manitoba candidate Thomas McDonald by ninety votes. He identified himself as an independent candidate during this campaign, although most political observers considered him to be a Conservative in all but name. He appears to have rejoined the Conservative caucus during the parliament that followed. In the 1927 election, he lost the Ste. Rose constituency to Progressive candidate Maurice Dane McCarthy by 224 votes.