How does Mike’s reaction to the swim demonstrate his personality?

Are you asking about Mike’s reaction to the swim team as a school organization, or to the individual swim contest he takes on against Chris Coughlin? We can look at both, because his reactions are similar.
Mike Barbour is the star linebacker on the football team at Cutter High School. He is a major protagonist to the main character, The Tao Jones, known as T. J. Mike also demonstrates the behavior of an arrogant bully, both verbally and physically, whenever he feels as if his semblance of personal power is being challenged. T. J. makes a habit of deliberately challenging Mike, since few others will do it. T. J. thinks his best way of putting Mike in his place will be to run a successful swim team whose members earn the prestigious school letter jackets.
After the team’s first swim meet, Mike confronts T. J. (near the beginning of chapter 9). “What was that, some Special Olympics swim meet?” he asks. “You’ll never see one of those goofballs you call swimmers in a Cutter letter jacket,” he says. And he uses his influence with the Athletic Council to change the requirements that the swimmers have to meet in order to earn the jackets.
The individual swim contest is arranged during the Athletic Council meeting near the end of Chapter 14. When T. J. hesitates a bit after making the offer, Mike immediately pounces by saying, “What’s the matter, tough guy? Open your mouth a little too quick?” But no, the contest between Mike and Chris stands. It takes place early in Chapter 15. Mike obviously thinks he’s going to beat Chris because he believes he is the better athlete and is smarter, too. We aren’t shown the background, but we can assume that Mike has done no extra preparation for this match. On the other hand, Chris has been swimming every day for three months and is used to the interval training technique. He has stamina. When Mike is told the rules, he quips, “Let’s just do it.” He “flies out over the water with a grunt,” to start. But this is not a race, it is a test of endurance. Mike loses after just four hundred yards. Instead of congratulating the winner in the spirit of a fair athletic contest, he says, “Get away from me, you little retard.” And he aims a put-down to T. J. too, saying, “You tanked that last race. You’re the only guy who doesn’t letter.”
Mike shows his bullying personality in both cases. Instead of supporting these budding athletes on their new team and their quests for jackets, he puts them down and insults them. He probably thinks that by doing so, he builds himself up in turn, at least in his own eyes and in the minds of his followers. He has to be the stereotypical “big man on campus.” He has some serious introspection to do in order to deal with his own demons. Maybe he learns something by the end of the book.