Qualifications
Well, a law degree helps, although it isn't a requirement. So does the natural ability to get along with difficult people (these two often go hand in hand).
More than anything though, it's all about leadership skills, and your powers of persuasion. You get yourself into the position of union head by proving that you can make corporate types see the light, and turn them around to your way of thinking. You can't be shy, or hesitant, or exhibit any lack of confidence or passion in what you're fighting for. You have to be the guy or gal in which everyone else feels comfortable placing their faith.
You're like the captain of a football team. You may not run the fastest forty or be the most physically impressive specimen, but your teammates turn to you and believe in you. They know you can get the job done—that you're the one to lead them into battle.