You have this tagged with "end of the Cold War" so I am going to assume that you are talking about the global community that was emerging in the early 1990s.
If so, I would say that the United States played the role of a global hegemon and sole superpower during this time. In a large way, it continues to play that role today, although China is starting to compete to some extent for superpower status.
The status of the US could be found, for example, in the way that it organized a coalition to fight against Iraq in the first Gulf War of 1991. In addition, the US led efforts to stop or prevent disorder in such places as Somalia and various parts of the former Yugoslavia.
Global community implies a world in which barriers and differences between people based on nationalities are diminishing. These barriers have not been lowered because of planned attempts of any big and powerful countries to create a global community. The global community has emerged as a result of evolving technology and the resultant increase in global trade and interaction.
Global community has emerged because people exposed to a wide range of concepts, and cultures have accepted and adopted what they liked best, irrespective of the the source of such practices. This has enabled people to choose what is best suited or liked by them irrespective of the constraints of their own community and culture. This has enriched lives of individuals, and at the same time developed better understanding among people of different countries.
What the U.S. or any other country can do to promote global community is to not to encourage any actions that tend to isolate their people from rest of the world by creating a false sense of their superiority over others. Development of global community is because of the free choice given to people by the technology. The best way to speed up this process is to not to create new barriers between people.