Robb Stark is a little too busy being King of the North and fighting against the Lannisters to spend much time in A Clash of Kings, but we do get a few moments with him to see how his character has changed since the first volume.
The eldest Stark may be the King of the North, but he is still stuck in that awkward phase between being a boy and being a man. His "beard [has] grown in redder than his auburn hair" and he seems to "think it [makes] him look fierce, royal… older" (8.Catelyn.49). But he is still only fifteen years old, and we're pretty sure Catelyn is being generous by referring to that peach fuzz as a beard.
Even so, Catelyn does note that either the war or the crown has forced him to grow up fast (8.Catelyn.58). He wins several victories against the Lannisters, including a battle in which Ser Stafford Lannister is killed—and the victory is so complete that Ser Lancel claims Robb must have used "'vile sorcery'" (33.Sansa.14). Go Robb, we suppose.
Robb also tries his hand at playing the game of thrones. Telling his mother he'll send the boisterous Greatjon Umber to parley with Renly, Catelyn has no choice but to agree to go as the envoy instead. Sure, Catelyn sees through the ploy, but even she admits that it was "deft for a boy of fifteen" (23.Catelyn.20). Mad props from his mama.
Robb spends a good portion of the second half of the story on campaign and away from any point-of-view characters. As such, we'll have to wait and see how much the war and responsibilities of the crown change him further.