After the Cuban Revolution, Cuba (or Fidel Castro) tried to rebuild the government by creating a vanguard communist party that would serve as the effective leaders of the country.
Before and during the revolution, there had been many different groups that had opposed Batista. After the revolution, Castro unified these groups under his own leadership. By 1965, these organizations had been become the Communist Party of Cuba. This party was the real government of Cuba.
So, the basic process was one of amalgamation of the revolutionary groups that had opposed Batista. These groups were combined into a communist party which took control over all major groups in society such as labor unions. It used its control over these groups to give it enough power that it essentially became the government of Cuba.