For a team to be successful, the teammates have to know that they will look out for one another. When a team member cares about no one but himself, the whole team suffers. I have found that one of the best ways to get members of a team to care about one another is to get them together outside of a work context in order to build relationships.
Plan retreats and other events that put people together in social settings. And during these times, make sure that they spend part of their time with staff members they don’t really know very well. That way they’re not only building relationships, but they’re being prevented from developing cliques.