What kind of boy is he? How does Haddon develop his character? How does he change and grow during the story? Do you feel sympathy for him -- if so, how does Haddon encourage this? Is he capable of love as 'we' feel it? Does he feel emotion? What will happen to Christopher in the future?
Are there any terms for the character development I can use, literary elements, etc?
And what can you tell from the context about his mother?
You might talk about direct and indirect character development, discussing how much of Christopher's character is revealed directly through the narrator's comments (in this case of first person narrator, Christopher's own comments) and how much is revealed indirectly through his conversations with his neighbors and their comments to or about him.
We are able to feel sympathy for Christopher. By experiencing the story as Christopher does, we view the world through his eyes and are able to understand his world. I don't think this would have been possible if the story had been told about Christopher. It would be hard, I think, for people with no personal knowledge of Asperger's syndrome to understand why he had to do some of the things he did or why his rituals were so important. Christopher has to learn to read faces to interpret emotions and he has no concept of the emotions that others around him feel. He does suffer, I think, when he decides to leave his father. Christopher will find success in his future. He proves this when he ventures out and searches for his mother. This shows growth and a change in his character.
The primary vehicle which is used in this excellent story to develop the character of Christopher of the mode of narration. The first person narration allows us to enter the world of a teenage boy with aspergers and to see everything from his perspective. We understand the curious literal world that Christopher inhabits far better than if he was a character that was described to us.