As in most dramas, the characters are defined by their actions. Dr. Stockmann's relentless pursuit of his quest for truth shows him to a man of principle and possibly naïve idealism. The machinations of his brother, Mayor Stockmann, reveal the Mayor's ultimately crafty yet practical nature. Hovstad's betrayal of the Doctor demonstrates his hypocrisy. And then, of course, there's Aslaksen, whose timid nature is shown by his refusal to take any action that ruffles feathers.
Plays are written almost entirely in dialogue, so it makes sense that the characters that inhabit them are often defined by the way they speak. Dr. Stockmann tends to speak in high-energy bursts and has a tendency to go off on long monologues. This all seems to point to the idea that the Doctor is a guy who has a lot of ideas and is super enthusiastic about sharing them.
A distinct difference in personalities becomes pretty clear when you compare the way Dr. Stockmann talks to his wife. Rather than haranguing everybody with her ideas, she stays quiet most of the time, only occasionally butting in to try and reign in her husband's wilder ideas. Mrs. Stockmann's restraint in her speech denotes her cautious nature.
Social status is a big deal in An Enemy of the People. All the characters are defined by it in some way. The Mayor uses his top-dog status to assert his will over just about everybody. Hovstad uses his position as editor the paper to gain power and as a tool for his liberal political agenda. Then there's Dr. Stockmann, who status goes from respected man about town to lowest of the low. His journey as a character is clearly marked by his journey down the social ladder.