Character Analysis
A Man of Vision
We first meet Rilian as a character in his own story. When the parliament of owls tells the story of King Caspian's double loss (his wife's death and son's disappearance), they describe the trouble that came to Rilian after his mother's death and his search for the serpent: He "had a look in his eyes as of a man who has seen visions…" (4.59). This is, in fact, not a good thing. Basically, Rilian looks like a man who is out of his mind.
This is a condition that will persist with him for the next ten years. On the giants' bridge, Rilian appears as the silent Black Knight, a blank slate that gives Puddleglum the creeps. (What if you raised his visor and nothing was there? Eek.) When Jill and the others encounter him in Underland, they notice that "something about his face didn't seem quite right" (10.151), and after he begins talking in ridiculous and disagreeable ways, Jill wonders if he isn't a bit "silly." In both cases, no one thinks about enchantment.
A Chip Off the Old Block
Even when Rilian is enchanted and being obnoxious, there's still a pleasing quality about him that can't be suppressed. When he greets them in Underland, the children and Puddleglum see "A young man with fair hair […] He was handsome and looked both bold and kind" (10.151). After he is un-enchanted, Rilian begins to think and act like a true royal of Narnia—in other words, just like his father.
When the plain, black shield he's carried as the Black Knight is emblazoned with the red lion of Narnia, Rilian warms to his noble nature: "Doubtless […] this signifies that Aslan will be our good lord, whether he means us to live or die. And all's one, for that. Now, by my counsel, we shall all kneel and kiss his likeness" (13.191). It's clear that Rilian has inherited his father's devotion to Aslan and his moral uprightness.
But he's also inherited Caspian's wanderlust, for good or bad. Although he is intent on returning to Narnia for the sake of his father and his subjects, Golg the gnome makes him an offer that he almost can't refuse: a side-trip to view the wonders of Bism, the fiery city at the bottom of the world.
Caspian has an almost identical decision to make at the end of The Voyage of the Dawn Treader, when he is tempted to remain with Aslan and make no return to his kingdom. Rilian recalls his father's great adventure and makes it an argument for pressing on to Bism: "My father went to the world's end […] It would be a marvelous thing if his son went to the bottom of the world" (14.207). But Puddleglum successfully reminds Rilian of his duty, and the prince—like his father before him—makes the decision to go back. Phew.
It's no surprise, then, that after mourning for his father, Rilian goes on to rule Narnia well and bring happiness to all of his subjects. He clearly understands that a good king always puts the needs of his kingdom and people ahead of personal desires.
Prince Rilian's Timeline