As the protagonist, Christopher is the character we most expect to grow and develop in the course of the novel. However, it's sometimes difficult to measure how much Christopher has matured as a character, if only because his thought processes are so different from other people.
The subtleties of Christopher's inner development - that is, his emotional growth - are not as evident as they would with other first person narrators, in part because he explicitly denies the importance of this aspect. He states time and again that the brain is as mechanistic as a computer and that humans are no more special than animals - views that reflect his own experiences with the limitations from autism. For him, the seams of human thought processes show up more obviously than others, and he has accepted them. Further, his social skills are limited by his autism, which makes it easier for him to identify with animals than people, which is why he values them as much as humans.
His emotional expressions are expressed in a limited fashion, and for a simple reason: when they grow too strong - such as when he discovers his mother is alive or when he travels to London by himself - it manifests in a more physical manner, such as growing sick or blacking out or his mind overloading. Add to this, his emotional reactions to certain revelations are limited by his focused perspective: he is not upset to find out of his mother's affair because he assumes his mother is dead; he does not appreciate the significance of almost dying at the train tracks of London because he was too busy fetching Toby. However, he does feel loneliness and heartache as he leaves Swindon for London, and there is a note of personal triumph at the end of the book which is different from what we encounter before. In this sense, his character is allowed to feel new emotions through the experiences he goes through.
The most obvious measure of growth are the accomplishments he lists in the last paragraph of the novel, the observable events which make up the bulk of his story. Christopher uses these events as proof that he can fulfill his ambitions to go to university and be a scientist - for him, there is an irrefutably logical chain between accomplishing the past set of actions and fulfilling the future set of actions. He has become a more confident and experienced person, better able to cope with the complexities of life after his unusual adventures.
The development of Ed Boone as a character is interesting in how he starts as an apparent saint and becomes more flawed as the story progresses. We are introduced to him as a patient father and widower, but flashes of anger at Christopher's mystery investigation show that there is more to his personality than infinite patience and kindness. The depths of his patience, however, are upended by the secret he keeps: if anything, the decision to tell Christopher that his mother is dead shows a lack of patience, a search for the easiest, most convenient way to cut ties with Judy Boone and spare Christopher any suffering. However, Ed Boone's innate goodness resurfaces as the story comes to a close. His understanding of Christopher, called into question by the secrets he kept and the anger he vents as he loses control of the situation, is also regained in the project he describes to his son about rebuilding trust. He does not see the trust as being recovered immediately and even states he'll work at it for as long as it takes, which shows the reserves of patience which readers first associated with him does indeed exist.
In contrast, Judy Boone's character undergoes a more straightforward positive transformation. From the start, we know she didn't have as much patience with Christopher as Ed; however, the extent of her inability to handle him only becomes clear when we discover she isn't dead but has run away with Roger Shears. She is traumatized by the news that her son believed her dead for the past two years; though this is never stated as a motivation for reclaiming her responsibility as Christopher's mother, it does tie into a symbolic "resurrection" for Judy Boone. She not only comes back from the dead in Christopher's eyes, she also comes back reborn as a better parent and more committed caregiver, even giving up her relationship with Roger Shears to take care of her son. A line about being prescribed medicine that keeps her from being sad indicates the possibility that she suffers from her own mental illness - depression, perhaps, though the scattered thoughts of her letters may indicate something more serious. At the end of the novel, she seems as stable a force in Christopher's life as Ed Boone.
Outside of the Boone family, there is no discernible character development. Indeed, we do not know if there are any further resolutions in the relationships with important characters such as Mrs. Alexander or the Shears. That is because the story is being told from Christopher's view and the restoration of the family - and his success at his math exam - are the most important concerns by the end of the narrative.