After a long rule of the conservative Union Nationale, the Liberals won the election in 1960, inaugurating a period called the Quiet Revolution. During that time, the State grew and several institutions were created, encouraging a feeling of nationhood in the Province of Quebec. This was combined with the global 60s context, in which several separatist and decolonization movements emerged around the world, along with many left-wing movements in European countries. You had the Cuban revolution, the FLN in Algeria, the PLO in Palestine, the Viet Minh and Viet Cong in Vietnam, and so on. In First World countries, you had the Brigate Rosse in Italy, the Rote Armee Fraktion in Germany, the Weathermen in the US, the Provisional IRA in Northern Ireland...
During that decade, Quebec's left-wing became increasingly critical of the Liberal Party and its Quiet Revolution, deemed as too "quiet" and not "revolutionary" enough. This discontent was even greater when the Union Nationale made a comeback by winning the 1966 election. Meanwhile, separatists parties such as the conservative Aliance Laurentienne (founded in 1957) and the left-wing RIN (founded in 1960) were formed, while their mainstream appeal grew during the decade.
The FLQ was created in that context by young radicals who were feeling the political game was rigged by the "colonizing" power of Canada, and that armed struggle was the way to wake up the sleeping masses. You have to remember that, although improving at a light-speed rate, life conditions of French Canadians were still far inferior the those of English Canadians. Many felt that institutional racism was preventing French Canadians from obtaining meaningful jobs in public service or big industries, and many at that time compared their situation to the one of African Americans (in fact they fared worse on many indicators including mean revenue, level of education, literacy and child mortality).
Terrorist attacks began in 1963, aimed mostly at rich English neighborhoods, banks, English-owned business and the Canadian military. We see about three waves or groups naming themselves the FLQ at that time, most of them being caught by the RCMP after a few months of activity. By the middle of the 1960s, a new and more ideologically sophisticated group emerged under the guard of Charles Gagnon and Pierre Vallières, two intellectuals who will be arrested while protesting for a free Quebec in front of the United Nations in New York in 1966. Before that, the two had tried to establish political contacts with the Black Panthers in order to get weapons. It is during that incarceration that Vallière wrote Nègre blancs d'Amérique - White N*** of America -, a book that would become an unofficial manifesto for the FLQ.
Meanwhile, the mainstream started sympathizing with the FLQ, as many prominent artists were getting involved in the fight for "political prisoners", and social unrest became common, such as in movements like McGill Français or the 1968 St. John riot. It led a new generation of young enthusiasts to join the fight and launch their FLQ cells.
During that time, a popular liberal minister and ex-journalist, René Lévesque, defected to form the Mouvement Souveraineté Association (1967). His ecumenical party soon joined with the right-wing Rassemblement National and the left-wing RIN to form the Parti Québécois in 1968. This new party was meant to be a political and pacific alternative to terrorism, but it didn't fare very well in the 1970 election, winning only 7 seats.
While everything was moving fast in the province, the young radicals of the FLQ thought it was not moving fast enough, and they decided to take action against the status quo and the newly elected Liberal government of Robert Bourassa. It would culminated in October 1970 by the kidnapping of British commercial delegate James Cross - a symbolic move against British imperialism - and of Labour minister Pierre Laporte - a symbolic move against the "collaborating" Quebec government. As for what happened after that, interpretations differs.
Some Quebec nationalists (even serious historians) go as far as conspiracy theories, claiming that it was all a set up to crush the independence movement. Others are more nuanced. Still, on October 15th, a few thousand young Québécois went to the Paul-Sauvé center, chanting "Québec Libre!" and "Vive le FLQ!". The morning after, about 500 partisans - most of them only separatists, not terrorists - would be rounded up by the RCMP and Montreal Police under the "War Measures Act" while the army came in to protect sensible targets. Laporte would be "executed" the day after, probably while trying to escape. The discovery of the body would then plague the FLQ forever in the court of public opinion, as most separatists would turn to the non-violent PQ after that. The party would end up winning the 1976 election in a landslide.