This is a difficult one, as the team dynamics specifically depend on the individuals included. Though is not a single "fix all" answer, there are absolutely many ways to build structure and team cohesiveness. The first thing I would say, is to be an effective leader, you must have the ability to identify and really understand not only the strengths of your teammates, but also their weaknesses.
Knowing the weaknesses and being able to somewhat remedy them by building on the strengths is how a leader, or in your case a team captain improve overall performance, but more importantly, unknowingly gain respect from the others.
Instead of focusing on what you cannot do, try to focus on what you can. Try a some internal competition. There's nothing wrong with friendly competition, as long as the end goal is for the better of the whole team. Pair people up, strengths against weaknesses. In many cases, as long as the person has a truly competitive spirit, this will push one toward striving more to turn their weaknesses into strengths.
Don't wait for the coaching staff to crate the fun environment. Everyone has the power to be creative. Use that to build a friendly yet competitive atmosphere within the team itself. Like other sports, true athletes strive to be the very best. Sometimes competing with your peers creates a tighter bond, thus a more competitive and supportive dynamic when it comes to the real deal.
Sometimes leading, is really doing so without people realizing they are being led. Try getting the rest of the team involved in competitions with each other. It is important that you are also part of it, as respect comes from doing. Don't worry so much about possibly being beat. You won't lose them if you keep up the motivation.
Leaders so often suffer from the fear of losing, thinking that if they lose to a subordinate it will cost them their position. Don't think of leading as a position. Think of it as a privilege to inspire.