The Americans fought in Vietnam—beginning with a handful of advisers sent in 1950 by President Harry Truman. But the Vietnamese have been fighting occupation for millennium. Trace the "roots" of the American war in Vietnam. Was this, from the Vietnamese perspective, basically an anti-colonial war?

From the perspective of most Vietnamese, this was a nationalist struggle to kick out foreign occupiers. Vietnamese are taught today to think of leaders who fought against Chinese occupation as early nationalist heroes. When the French invaded under Napoleon III, that struggle then turned against them. When Americans then invaded (and actually, Eisenhower had subsidized the French) most Vietnamese again saw this more as a national struggle.
There were huge exceptions, of course. Communists, a minority in the Vietnamese coalition to kick out the Americans, ironically became stronger the longer the US stayed in Vietnam. There was a long series of Vietnamese collaborators with foreign occupiers, Vietnamese Catholics, mixed French Vietnamese people, and people of Chinese people who made up much of the merchant class. Finally, there were over a dozen Hill Tribes who had fought against the lowland Viet people for centuries.

Before delving into this question in more detail, I think it's important to think about which perspective you want to understand. Please note that there were two Vietnamese perspectives regarding the nation's destiny after the people, under Ho Chi Minh, declared independence from the French on September 2, 1945, establishing the Democratic Republic of Vietnam.
The brilliance of Ho Chi Minh is that he was a great diplomat and a man who knew how to appeal to very different figures in power to help ensure freedom for his people. Thus, during the Paris Peace Conference in 1919 when the Treaty of Versailles was drawn, he appealed to President Woodrow Wilson for American support in freeing the Vietnamese from colonial rule. Ho was using one of Wilson's terms in the Fourteen Points regarding the release of colonial holdings, an effort that signaled the beginning of the end of the Age of Imperialism.
During World War II, Ho collaborated with the Allies against the Japanese, who occupied French Indochina in 1940, and supported French colonialists associated with the fascist Vichy regime. When Japan surrendered in 1945, the French seized South Vietnam and opened talks with the Communists in the north. Talks collapsed when French warships bombarded the northern city of Haiphong, killing thousands.
During the post-war period, predictably, the French and the Americans supported the anti-Communists in the South, while the Chinese leader Mao Zedong supported the Communists—later known as the Viet Cong—in the North. Ho Chi Minh's wish, which was followed by the Viet Cong guerrilla fighters, was to reunite his country under Communist rule. He wanted Vietnam to be free of all Western interference. In that respect, the Vietnam War was, indeed, an anti-colonial war. The Viet Cong wanted to expel both the French—who represented the old colonialism of economic exploitation—and the Americans who represented the neo-colonialist tendency of directing governments to behave in their interests.
The South Vietnamese supposedly wanted independence, too, and claimed to model themselves on Western democracy. However, suppression was common. It should also be noted that there was a great deal of corruption and infighting between South Vietnamese leaders. Dissenters were jailed. This isn't exactly what democracy looks like. One wonders, too, what kind of government would have taken over in Vietnam if the South Vietnamese had won. Might it have been a U.S.-allied dictatorship along the lines of Ferdinand Marcos's presidency in the Philippines, or Suharto's in Indonesia?
After the fall of Saigon on April 30, 1975, the capital of South Vietnam—Saigon—was renamed Ho Chi Minh City, representing the fulfillment of the leader's dream.
https://www.history.com/this-day-in-history/vietnam-independence-proclaimed