Arya Stark is, without a doubt, one of the most important and powerful characters on Game of Thrones. She might not have any claims to the Iron Throne, nor a battalion behind her, but she is the best fighter in Westeros, and she also happens to be the only one able to kill the Night King, making her the savior of all of humanity. So it makes sense that where is Arya going at the end of Episode 4 is somewhere she feels she needs to be. In fact, at the end of the episode, "The Last of the Starks," Arya is heading to King's Landing with the Hound, where both have some unfinished business to attend to.
After Arya killed the Night King in the Battle of Winterfell, she's in a sort of resting, in-between moment, but that doesn't mean she's resting. In fact, while everybody else is celebrating the North's victory over the White Walkers, Arya is practicing her archery skills alone. No rest for those with murder lists.
Meanwhile, after all the events of the last few episodes, The Hound is leaving Winterfell, ready to move on to King's Landing after the big battle. Towards the end of the episode, Arya caught up with him, and decided to join him on his trip. They both like to be alone, but they can be alone together.
The Hound doesn't say why, exactly, he's on his way to King's Landing, but Arya knows the grudge he holds against his brother, The Mountain, who stands by Cersei's side. Fans have been dying for Cleganebowl even though it seemed like a pipe dream up until now, but with this duo moving south, it could be a real possibility.
Even though The Hound was once on Arya's list, she took him off of it mostly because she thought he was dead, but as this scene shows, they've also come to foster a kind of mutual respect. They're kind of alike, and now are both heading to King's Landing to impose justice upon their enemies: The Mountain and Cersei Lannister.
The Mountain is one of a few names still on Arya's kill list, along with the current Queen of Westeros. He's also a zombie-like creature made to protect Cersei at all times. It's possible that Cleganebowl could happen, but Arya will be the one to take the kill, and subsequently, his face, which would help her kill Cersei. There are also theories that Arya will take Jaime's life and face to fulfill Maggy The Frog's prophecy of the "little brother," or valonqar, choking the life out of Cersei. (This is also very possible, as Jaime is heading south to King's Landing as well after initially saying he'd stay in Winterfell.)
Either way, that's where Arya is heading: to King's Landing to end Cersei's life. And, as revealed in her conversation with the hound, after she kills the queen, she doesn't plan on coming back to Winterfell. Could she be planning on dying in her quest to kill Cersei? It's definitely possible. She also might think that her skills are needed elsewhere, and there's nothing left for her at Winterfell, which is also possible. Perhaps she is planning to head back to Braavos to become a full-time assassin with the Faceless Men.
With Arya heading south and the Battle of King's Landing expected to start in episode 5, it's only a matter of time before more key players are killed, Cersei being one of them.