Why did Hawthorne use a minister in this story? Why didn't he use a different character like a baker or a teacher?

What a wonderful question!  My opinion of Hawthorne's choice of his main character's occupation is that the minister, more than a baker, teacher, or any other secular occupation epitomizes morality.   Ministers, more than anyone else (with the exception of nuns, priests, bishops, and other members of clergy) are expected to be pure, clean, and lead lives worthy of imitation by the rest of us sinners.  That's not to say that bakers and teachers aren't moral or lead clean and pure lives, but their occupation titles don't necessarily lend to all the rest of society expecting them to be "angels".

Using a minister for this story puts emphasis on his life.  His purpose in wearing the veil is to show his congregation that we all have sin...even ministers...and while it ruins his life in terms of his relationship with Elizabeth, it does improve his ability to be effective as a minister.  His congregation come to him more often with their confessions and problems than they did before because he appears to them more human, more like them.  Perhaps the veil gives him accessibility...an air of trust that perhaps they didn't have for him before he began wearing it.