when did he become the prime minister?
June 30, 1984.
John Turner
Turner then re-entered politics, and defeated Jean Chrétien, his successor as finance minister, on the second ballot of the June 1984 Liberal leadership convention. He was formally appointed prime minister on June 30. 
who is john turner? Canadian politician and lawyer what was his political position? Served as 17th prime minister of Canada, the cabinet of Prime Minister, and contested the 1984 Liberal leadership.
wiki:5404914
John Turner [SEP] Prime minister (June–September 1984)
Trudeau retired after polls showed the Liberals faced certain defeat in the next election if he remained in office. Turner then re-entered politics, and defeated Jean Chrétien, his successor as finance minister, on the second ballot of the June 1984 Liberal leadership convention. He was formally appointed prime minister on June 30. When he was sworn in, Turner was not an MP or senator. Had he wished to have parliament summoned, he would not have been able to appear on the floor of the House of Commons. He also announced that he would not run in a by-election to get into the Commons, but would instead run in the next general election as the Liberal candidate in the British Columbia riding of Vancouver Quadra.