EDIT: I feel I should make clear that while my area of expertise is the Vietnam War, I do not command expert-level knowledge on Buddhism. Therefore, while I can explain the events leading up to Thich Quang Duc's self immolation, I am unable to explain the roots of that practice in detail.

Catholicism in Vietnam indeed stems from French influence. French missionaries began traveling to Vietnam in the 1600s. The Vietnamese monarchy generally saw Catholicism as a potential threat, and so religious persecution was high for many years. This persecution was one of the main justification France used to begin annexing Vietnam. The ruling Catholic minority in 1963 was Ngo Dinh Diem's government. French colonial rule in Vietnam ended in 1954 when the Viet Minh, an anti colonial group spearheaded by the Indochinese Communist Party, defeated France in the First Indochina War. The US had supported France against the Communist forces throughout much of this conflict. After France's defeat, the US established its own presence in South Vietnam, which was a creation of the peace agreements that ended the war. Under this treaty Vietnam was partitioned in two - the Communists got the North and all other non-Communist groups got the South. Diem was a staunch Catholic as well as Vietnamese patriot who had never collaborated with the French. As such, the Americans considered him untainted - almost everyone else they could have put in power had collaborated in some way, and they didn't want the South Vietnamese government to be associated with colonialism. His Catholicism came into play because some policy makers thought Catholics more trustworthy and anti communist than Buddhists. Buddhism tends to stress neutrality and pacifism, while the Catholic Church had been strongly anti communist for decades.

After being elevated to the office of Prime Minister of the State of Vietnam (South Vietnam's official name in 1954) Diem promptly organized a rigged election in which he transformed the State of Vietnam into the Republic of Vietnam, with him as President. In the process he kicked out the last official emperor of Vietnam, Bao Dai. Once in power Diem stacked government positions with his family members and refugee Catholics from the North. He also cracked down on all rival groups, in some cases having popular political or religious figures executed or exiled. All of this created an environment in which many Buddhist monks felt discriminated against. It came to a head in mid 1963, when several monks in the central Vietnamese city of Hue asked permission to fly the Buddhist flag on Buddha's birthday. Under RVN law, only official government flags could be flown outside. This law did not stop Catholic churches from flying the Vatican flag, so the monks did not foresee any problems. But Diem's government denied this request. In response, the monks organized a protest, at which a bomb exploded and killed several people. Diem blamed the incident on the Communists, but the Buddhists blamed his government, and immediately began organizing wider protests against his autocratic policies. Vietnamese Buddhist monks had long practiced self immolation as a means of honoring Buddha, and when Thich Quang Duc did so, he was surrounded by Western reporters, which caused the crisis to become international news.