Why is Jon Snow traveling north of the Wall?
Jon was always the show's most clearly defined hero, and while Jaime's regicide polluted his reputation and self-identity, it doesn't seem to linger on Jon in the same way. Similarly, while whatever was left of Daenerys' heroic legacy disappeared the instant she went crazy on King's Landing, Jon is not negatively affected by his betrayal of her. He lives with the guilt of his actions, but gets to go back to a familiar place and do...well, what, exactly?
Without the White Walker threat -- or any Wildling fear that the "Crows" had at the beginning of the series -- what purpose does the Night's Watch serve? Honestly, it's unclear, though Tyrion says bastards and broken men always need a home. Whatever it's purpose, it seems like Jon is not long for that organization, or what's left of it, anyway. His "sentence" is merely a way for competing interests left alive to find a compromise. As Tyrion tells an imprisoned Jon, "Releasing you to the Unsullied would start a war. Letting you walk free would start a war. So our new king has chosen to send you to the Night's Watch." Basically, Jon's being told to go as far away as possible, and he can do whatever he wants as long as he doesn't start a war in King's Landing.
The final shots of the entire show find Jon, Tormund, and Ghost leading a crew of Wildlings beyond the Wall. As the gate to Castle Black closes behind him, Jon gives it a forlorn glance. This time he seems to be leaving that world behind for good. Is Jon going searching to make sure all the White Walkers are for sure dead? Is he setting out to take up Mance Rayder's old title as the King-Beyond-The-Wall? That latter option seems like the most likely. Still, Jon's story is open-ended. Almost all the other living characters are settled into prescribed roles, but Jon Snow could go anywhere. It's a moment that winks at the notion that perhaps his story is not totally done in that it screams spin-off potential, but it also reinforces who showrunners David Benioff and D.B. Weiss always considered the true protagonist of this saga. Jon's actions in the finale may be morally suspect, but Jon himself is not. He gets his version of a happy ending.