For the explorer and the officer, the two main characters, this is the most important characterization tool. Each of them has a very different set of beliefs, significantly at odds with each other, and much of our understanding of each comes from their beliefs. For most of the story, the officer's the one doing most of the talking. He describes the old Commandant, his judicial procedure, and the apparatus, and from his enthusiasm for these things, we learn a lot about him – he's passionate, he seems to have a weird thing for pain and torture, he believes that one should never doubt guilt, and he lives for his devotion to "Justice" as he sees it. The explorer's opinions we learn more from his reactions, whether spoken or thought.
There aren't a lot of big or dramatic actions in the book, except of course the officer's decision to go under the apparatus, which shows better than any words just how bound up he is with his devotion to the world of the old Commandant. However, smaller, less exciting actions define the characters fairly well, especially the minor characters. The soldier is a bit of a good-for-nothing who falls asleep and is incompetent all the time. The condemned man plays with cogwheels like a curious animal might, creeps up gleefully to watch the officer die in the machine out of his vengefulness, and displays his submissiveness by getting on his knees when the explorer tells him to stop watching. The officer's love of the old Commandant and the apparatus is apparent in how carefully he handles it.
This basically only applies to the condemned man, who is described at the start of the story as a "stupid-looking, wide-mouthed creature with bewildered hair and face" and, shortly thereafter, as looking "so like a submissive dog" (1). That sums up his character pretty well.