Currently, I am in the process of learning Chinese. As such, I have taken to reading children's books, since they are the only ones I can understand. In the process of milling through the text, trying to understand every word, I have put a lot more thought into the characters. If I was reading this book in English, I would never do such a thing.
The Cat in the Hat features seven characters: the Narrator and his sister Sally, the Fish, the title Cat, Thing One and Thing Two and the children's mother.
The pair of Sally and the Narrator are observers of the story, although the Cat's stated intention is simply to cheer them up on a rainy day, they never seem to derive much pleasure from what the cat does. They also don't show that much alarm at the cat's tricks, although they do seem to spend a lot of time staring at him dumbfounded. The narrator only speaks, and acts, once in the story, to talk about getting his net, and then to use his net to capture the Things. Perhaps Dr. Suess was trying to inject some realism into the story, after all, even at the age of five or six, most children would be a little shocked at the appearearnce of an anthropomorphic cat in their home.
Thing One and Thing Two are two small creatures of indeterminate type. They are the ones responsible for flying kites through the house, wildly knocking things over. I think perhaps the reason for their introduction is that, although he (she?) has very few feline traits, the Cat in the Hat still has one, that is, he is somewhat aloof. He is too mature to simply run through the house, knocking things over. Thus, he introduces the Things, who do not speak, to do this work.
Mother, who only shows up at the end of the book, and then only to see her foot coming through the door, is notable mostly for her absence. Her return, however, returns the story to reality.
This brings us to the two major characters in the story: the obvious Cat, and the often overlooked Fish.
The Cat in the Hat is the title character. The first thing to be noted about him is his overall felinity, or lack of it. While he does share some feline traits, such as the aforementioned aloofness and a certain playfulness, he is very obviously non-feline in some ways, such as the fact that he is a biped and much larger then the main characters. Psychologically, although he is manipulative and somewhat coy, as cats have been known to be, he is not shy or retiring. And overall, he lacks the one trait that makes cats cats, cuteness. The Cat in the Hat is simply not cute, physically or mentally. Instead of drawing towards him, and wanting to play with him, as children would be expected to do, the children seem to be repulsed by him, shrinking away.
The other question is whether the Cat is a benevelent trickster figure, or rather a cthonic force of nature, bent on hurting people. While in many trickster stories (such as the modern One Flew Over The Cuckoo's Nest), the plot of the story revolves around the trickster moving from being a simple minded, selfish michief maker to a constructive figur of independence, the Cat in the Hat doesn't seem to want to teach the children any lesson, and his mischief making seems either chaotic or cruel. It is true that at the end of the story, he returns with a machine to clean up the mess. However, this seems to be somewhat of a coda thrown on to the story, to prevent a sad ending where the children would have to explain to their mother what exactly happened.
The Fish is not at first an obvious character. However, discounting the narrator and the mother, he is the only character besides the Cat who speaks very substantially. He is the voice of reason, reminding the chlidren again and again that they shouldn't let the Cat in the Hat be in the house. His place inside the story is well suited for a fish (albeit a talking one). He can speak and warn the chlidren, but since he is very small and waterbound, he can not activly do anything about fighting the cat off.
As the question with the cat is whether he is a benevolent or cruel trickster, the question with the fish is whether he is genuinly interested in protecting the children, or is simply a self-righteous killjoy. Especially in Chinese, his cries of "Bu keyi! Bu keyi!" come across as the cries of someone who is trapped and bored themselves, and wishes others to share their fate. On the other hand, seeing the fish taunted and physically abused by the cat, it is hard to not feel a little sorry for him.
Of course, many of these issues (especially the one of whether the cat is dangerous or not) have been colored by the many changes in outlook that have occured since the book was first published, in 1957. Nowadays, a book about a character that seems to be an adult male entering into some children's house while they are alone and then doing what he likes, and leaving, seems a lot more threatening then it did in the days of the 1950's, when child abuse was not as widely acknowledged. Especially considering the fact that the children are unable to tell their mother was occured, their is a good chance that this story, with its creepy, threatening cat, could never be published if it was written today.