How did the Vietnamese communist party come to lead the independence movement?
I would point to two factors that allowed Ho Chi Minh and the communists to be the leaders of the independence movement after World War II.
First, before the war, the French brutally repressed any independence movement. This took away much of the support for any of the more moderate parts of the independence movement. The French repression made the communists, who were radicals and promoted violent revolution, seem more palatable to the Vietnamese people than the moderates who wanted to negotiate with the French. Second, the communists were the group that was most active in the fight against the Japanese during WWII. Because of this, they gained immense prestige in the eyes of the Vietnamese people.
So, because they were more radically anti-French before the war, and because they were the strongest anti-Japanese force during the war, the communists came to lead the independence movement.