How do the characters undergo change in the Harry Potter series along with the plot development?

It's worth noting, the Harry Potter series is a coming-of-age story, tracking the growth of its characters from childhood through adolescence and into adulthood. As the characters grow up, so do the books, becoming more serious in tone and much darker in theme. As the characters grow up, they find themselves dealing with the turbulence of adolescence, and by the end, the complexities of adulthood.
For one example, there is Harry Potter, himself, who begins as a neglected orphan. Even in those earliest books, he displays a great deal of raw bravery, such as when he opposes Quirrel in book one and fights the basilisk in book two. At the same time, however, he does mature emotionally as he learns more about his parents (and the complicated history between the Marauders and Snape), and experiences loss (particularly with the death of Sirius). All the while, there is still his connection with Voldemort, which plays a critical role throughout the series.
A second notable example would be Draco Malfoy, Harry's rival, who begins the series as a privileged and spoiled child, raised by a Death Eater family. Things change dramatically for him with Voldemort's resurrection, particularly in the events of the sixth book: his family loses favor and he is assigned an almost impossible task while the full weight of Voldemort's psychopathic displeasure weighs down on him.
The examples are innumerable, and practically all of the central characters receive character arcs that run throughout all seven books. There's not enough room to do justice to the full complexity of that picture which Rowling weaves in the bounds of a format like this.