The sigil of the Clegane family is three dogs on a field of yellow, and this sigil is very fitting for Ser Sandor, a.k.a. the Hound, and Ser Gregor, a.k.a. the Mountain. For these two brothers in arms, it's a dog-eat-dog world out there. And they are some pretty big dogs.
The Hound does not take the knight's vows because his brother is a knight and "he hate[s] his brother" (3.Sansa.32). But we think his dislike of knights goes a little deeper than this. The Hound is a very truthful fellow, and he feels that all the pomp and circumstance regarding knights' chivalry and honor gets in the way of the truth of the world. And the Hound's truth is survival of the fittest.
As he tells Sansa, "What do you think a knight is for, girl? You think it's all taking favors from ladies and looking fine in gold plate? Knights are for killing" (53.Sansa.46). He claims that as long as he has his sword he won't need to fear any man. In the Hound's world, it's kill or be killed—the strong survive because they devour the weak. Ned Stark lost his head because he was no longer the strongest, while Gregor Clegane can do what he wants because he is strong enough to do exactly that.
During the Battle of the Blackwater, the Hound becomes terrified of the fire, and he loses face when Tyrion takes the command to lead the offense. Realizing this, the Hound must leave King's Landing (63.Sansa.32). After all, if people say Tyrion is stronger than him, then he cannot survive in the same city.
If there is a soft spot in the Hound, it's Sansa Stark. It's never clear why he takes a liking to her: Perhaps it's because she's pretty, or perhaps it's because her suffering resonates with his own suffering at the hands of his brother. Whatever the case, the Hound never beats Sansa like the rest of the Kingsguard, and he protects her from the mob during the riots of King's Landing. He also affectionately refers to her as a "little bird." Despite his kindness, though, she can never look him in his scarred face.
His survival of the fittest worldview breaks down around her. When he decides to flee King's Landing, he goes to Sansa's room to ask her for a song. Being as strong as he is, he could easily force her to do anything he wants, but he doesn't; he asks her, and she sings for him.
Although it's too dark to see his face, Sansa does "lift her hand and cup his cheek with her fingers" (63.Sansa.51). Feeling him cry, Sansa watches the Hound leave.
Sandor Clegane may have a teeny-tiny soft spot in him, but his brother, the Mountain, is a pure force of nature.
In A Clash of Kings, Tywin Lannister lets Gregor off the leash to rampage across the Riverlands. He rapes, burns, pillages, murders, and otherwise terrorizes anybody in his path. Most of his exploits are given to us secondhand since so few point-of-view characters bump into him. Lucky them.
Brynden Tully notes that Gregor put an entire garrison to the sword, even Darry Raventree, the garrison's child lord, despite the fact that the boy would have made better ransom than victim. Brynden claims, "That beast's head would make a noble gift for all the people of the realm, I vow" (8.Catelyn.103). Emphasis on all the realm—there's not a lot the whole realm would agree upon, we're thinking.
Arya briefly encounters the Mountain when she and her group are captured by his men. The Mountain picks captives from the group and has his man, the Tickler, interrogate and torture them until they die. Throughout all of this, "Ser Gregor Clegane himself would stand motionless, watching and listening, until the victim died" (27.Arya.7). In other words, dude's not fazed.
We last see the Mountain riding into battle with Tywin Lannister's forces. No doubt we'll be hearing more about his violent career in the next volume.