So if there's one thing that the Hound loves in this world of Game of Thrones it's killing stuff, as far as I can tell he never gives a second thought as to whom he's fighting or why he's doing it. It is pretty clear that he has no interest in women (except maybe Sansa Stark), so I can't see any reason why he doesn't just go to the Wall. He seems to hate glory and fire, the two things that no one has to worry about in the Night's Watch. So I'm just wondering if maybe there is some obvious reason that I've missed for the Hound not going north.
Partly, it's miles away - not something southerners think about. He's far too cynical for solemn vows, and too much of an independent-minded loner to choose to huddle for warmth with hundreds of 'brothers'.
But mostly, it'd be unthinkable due to pride: Except for among the Starks (and a few other mostly-Northern families descended from the First Men, like the Royces and Mormonts), taking the black is associated with failure or disgrace. The only Southerners who go to the wall are sent there, and they're usually criminals, the defeated, or children deemed unworthy by their families (like Sam Tarly).
Sandor may be cynical, but he's proud, and has a complicated relationship with his brother based on loathing and rivalry. He takes pride in being a no-nonsense self-made man who achieved fame/infamy and genuine status off his own back, in contrast to his hated brother Gregor who was gifted a title of Knight and land despite (or, because of) being essentially a monster.
The books also show him taking pride in his family's history. The Cleganes earned their status through being good at dog breeding and handling, and he seems to see himself as similarly down-to-earth, honest and self-made - the most respectable and worthy Clegane, with contempt for all the phony, vain people around him.
It'd be unthinkable for him to take a position where he'd be the lowest of the low. He'd go from being the notorious Hound to just another failed reject, and his hated brother would be the most respected one in the family (and not just in the eyes of fools who think being a knight means something).