When we met Arya in A Game of Thrones, she was a plucky, nine-year-old tomboy who wasn't looking forward to a lady's life of dresses, dances, and quilting circles. She craved swords and sorcery, and most of all, a life of adventure. And, oh boy, did she get it. Unfortunately for her, it's a perilous time to pick up the adventuring hobby in Westeros.
Traveling north with Yoren and his Night's Watch recruits, Arya witnesses the horrors war has brought to the land. Bruised and bloodied pilgrims and road-side graves mark her travels north like something out of Dante's Inferno, and Arya gets firsthand experience with all the bad stuff the songs and stories gloss over.
She quickly learns that war doesn't play by the rules she's been told by stories and her father, Eddard Stark. When Ser Amory comes across Yoren's group, he commands they open the stronghold's gates to prove loyalty to King Joffrey, but Yoren refuses, likely knowing Amory won't let them live if he obeys. When Arya wonders why they can't see the Night's Watch's recruits are neither lords nor knights, Gendry answers, "I don't think they care, Arry" (15.Arya.71), and he is right: Ser Amory attacks and kills most of the group even though they pose no threat to him or his men.
Arya, Gendry, Hot Pie, and a few others then travel north alone, and things actually get worse. Consider the cuisine they dine upon: "She had broken her fast on some acorn paste and a handful of bugs. Bugs weren't so bad when you got used to them. Worms were worse, but still not as bad as the pain in your belly after days without food" (20.Arya.14). For a fantasy adventure, things certainly get real. Nothing's realer than starvation, after all.
And so, Arya learns that the world isn't a place for fun, wondrous adventures. The world is a cold, cruel place where the strong survival at the expense of the weak. And this knowledge begins to change her.
In fact, this knowledge changes Arya several times throughout the novel. Her identity crisis starts on the road with Yoren. The Night's Watch recruiter forces Arya to pretend she is a boy and go by the name Arry. Sure, it's not much of a stretch—we don't imagine Yoren was one with much imagination—but he does have a point that the road is safer for boys than girls.
Arya then adopts this strategy throughout her story, changing who she is and how she views herself in order to survive. When Arya gets to Harrenhal, the various dangers she faces cause her identity crisis go off the rails. Here's the identity crisis rundown:
As long as she is just surviving Harrenhal, and not taking charge of her own life, Arya's identity crisis continues. After hearing Winterfell has been destroyed, she wonders if she can still consider herself Arya or if she will become Nan the serving girl forever (65.Arya.69). Bummer, right?
But once Arya decides enough is enough, the crisis seems to resolve itself. Lost and confused, she prays to the old gods in the godswood and she hears (imagines?) her father's voice. Then she says:
"You are Arya of Winterfell, daughter of the north. You told me you could be strong. You have the wolf blood in you."
"The wolf blood." Arya remembered now. "I'll be as strong as Robb. I said I would." She took a deep breath, then lifted the broomstick in both hands and brought it down across her knee. It broke with a loud crack, and she threw the pieces aside. I am a direwolf, and done with wooden teeth. (65.Arya.104-5)
The wooden sword, a replacement for her lost Needle, is broken, signifying that Arya no longer needs substitutes. Instead, she remembers who she is and takes charge of her life; she becomes the genuine article rather than a self-substitute. With her true self intact, Arya escapes Harrenhal and decides to rejoin her family, her pack.
The thing about wolves is that they are predators. We may cheer that Arya has regained her sense of identity—of wolfhood let's say—but it comes with some seriously violent behavior, especially for a ten-year-old girl.
During her time at Harrenhal, Arya began saying her nightly prayers, but these are no "Now I lay me down to sleep" styled prayers. For this bedtime litany, Arya lists the people she wishes to kill, "polish[ing] her hates the way Gendry had once polished his horned helm" (27.Arya.24). And as her story continues, the list grows longer and longer.
Why this move toward violence? Consider all Arya has seen on her travels: She's watched the Mountain and his men kill without consequence; she's seen Jaqen H'ghar murder men, improving her life a bit by consequence; she even saw her father killed while Joffrey goes on unpunished as the King of the Seven Kingdoms.
Importantly, Arya hasn't just witnessed a ton of violence, she has also noticed that the men who perform violence don't just survive—they thrive. As such, she has begun to see violence, and the ability to commit it, as a path to empowerment, which is something she has lacked since she fled King's Landing.
This tendency toward violence reaches its peak during her escape from Harrenhal. Offering the guardsman a coin, she drops it. When the man bends over to pick it up, Arya "[slides] her dagger out and [draws] it across his throat, as smooth as summer silk" while reciting "'Valar morghulis'" (66.Arya.150-1). The man dies, and Arya, along with Gendry and Hot Pie, ride out of Harrenhal.
Now we have seen Arya kill before. In the previous volume, she killed a boy while escaping the Red Keep, but Arya was scared and her attack happened in the heat of the moment. Here, the murder is straight-up premeditated, a cold-blooded kill. Clearly, Arya not only sees violence as a path to empowerment, but one she is willing to take rather than be victimized by others.
By the way, valar morghulis means "All men must die" in High Valyrian, and Arya has taken a likening to the words (66.Arya.73). In her defense, she's seen them come true in spades.