Tools of Characterization
Characterization in The King's Speech
Social Status
Bertie is the King of England and Logue is a failed actor turned speech therapist. It's safe to say that the two of them live on different ends of the social ladder. And this difference in status tells us a lot about their personalities. Logue, for example, doesn't really acknowledge differences in social status because they prevent him from establishing trust with some of his clients.
Meanwhile, Bertie isn't too comfortable with getting emotionally close to someone so far outside his social class. It's only over the course of the movie that Bertie and Logue learn to move beyond their social classes and to become close friends.
Speech and Dialogue
In a movie about a speech impediment, it shouldn't be surprising that speech and dialogue are major markers of character. More specifically, Bertie's entire character has been touched by his stutter, which makes him self-conscious and short-tempered.
It's only by addressing his speech difficulties that Logue can address the other issues with his mind (and vice versa). Meanwhile, Logue has his own issues with dialogue, since people are constantly judging him for having an Australian accent and therefore not having a "pure" British accent like people from the upper classes of Britain.
Family Life
Logue is convinced that Bertie's family life has had some role in the stutter he developed when he was four or five years old. And he's probably right, because Bertie always seems to stutter more when he's around his father or older brother David, both of whom tease him about the way he speaks.
Also, Bertie has clearly been impacted by the death of one of his brothers in childhood. And that's not even to mention the fact that one of his first nannies hated and starved him. Anyone who didn't come out from such a messed-up childhood with some kind of issue would be super-human (or possibly a robot).
Names
When Logue first meets Bertie, he isn't interested in referring to him as "Your Majesty." He asks if he can call him by a friend's name, like "Bertie." But Bertie is super-uncomfortable with letting a stranger call him by the same name his closest family uses.
Over time, Bertie learns to accept Logue's friendship and he accepts that Logue calls him Bertie. By the end of the movie, even Bertie's wife Elizabeth is wiling to call Logue by his first name. And more importantly, both of them are happy to call Lionel by the name of "friend."