Sandor is wounded in the riverlands, and Arya leaves him there to die.
But is that it? What happens to him afterwards?
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We find out what happened to Sandor "The Hound" Clegane in a Brienne chapter in A Feast for Crows. We are first served some clues, then told outright. First we see Sandor's horse, Stranger:
Way down at the far end [of the stables], well away from the other animals, a huge black stallion trumpeted at the sound of their voices and kicked at the door of his stall.
Ser Hyle gave the big horse an admiring look as he was handing his reins to Brother Gilliam. "A handsome beast."
Brother Narbert sighed. "The Seven send us blessings, and the Seven send us trials. Handsome he might be, but Driftwood was surely whelped in hell. When we sought to harness him to a plow he kicked Brother Rawney and broke his shinbone in two places. We had hoped gelding might improve the beast's ill temper, but... Brother Gilliam, will you show them?"
[shows the horse bit off his ear]
Then we see Sandor himself:
...and higher still they passed a lichyard where a brother bigger than Brienne was struggling to dig a grave. From the way he moved, it was plain to see that he was lame. As he flung a spadeful of the stony soil over one shoulder, some chanced to spatter against their feet. "Be more watchful there," chided Brother Narbert. "Septon Meribald might have gotten a mouthful of dirt." The gravedigger lowered his head. When Dog [Septon Meribald's dog] went to sniff him he dropped his spade and scratched his ear.
And later on, clues of the Elder Brother's involvement with Sandor:
Unlike Septon Narbert, the Elder Brother did not seem dismayed by Brienne's sex, but his smile did flicker and fade when the septon told them why she and Ser Hyle had come. "I see," was all he said, ...
We get confirmation that the Elder Brother has met Sandor:
...There is one thing I do know, however. The man you hunt is dead."
That was another shock. "How did he die?"
"By the sword, as he had lived."
"You know this for a certainty?"
"I buried him myself. I can tell you where his grave lies, if you wish. I covered him with stones to keep the carrion eaters from digging up his flesh, and set his helm atop the cairn to mark his final resting place. That was a grievous error. Some other wayfarer found my marker and claimed it for himself. The man who raped and killed at Saltpans was not Sandor Clegane, though he may be as dangerous. [Note: It was Rorge]
I know little of this man, Sandor Clegane, ...
The brother claims that "The Hound" is dead, but when reading between the lines, it is clear that what he means is that Sandor Clegane has left his violent persona behind him.
... The Hound died there, in my arms. You may have seen a big black stallion in our stables. That was his warhorse, Stranger. A blasphemous name. We prefer to call him Driftwood, as he was found beside the river.
..."It is true, then." [Brienne] said dully. "Sandor Clegane is dead."
"He is at rest."
The Elder Brother then relates a story about himself being a knight, and being wounded on the Trident during the famous Battle of the Trident between Robert and Rhaegar, and finding his way to the Quiet Isle, where he found redemption. He finishes with:
"I see." Brienne did not know why he was telling her all of this, or what else she ought to say.
"Do you?" He leaned forward, his big hands on his knees. "If so, give up this quest of yours. The Hound is dead, and in any case he never had your Sansa Stark.
It is clear that Sandor lives, when reading this chapter. One can only hope that he does not stay there for a decade, like the Elder Brother. My personal hope is that he is reunited with one of the main characters, and becomes their guardian, preferably Sansa. Others have theorized that he will become a champion of the Light, and fight UnGregor in Cersei's Trial by Combat. Whatever the outcome, it is sure to be interesting.
Sandor's fate has not yet been confirmed.
Spoilers from A Feast for Crows!
However, there is a pretty solid theory going around that Sandor is alive and is a novice (gravedigger) at the monastery on the Quiet Isle. The evidence supporting this theory is as follows:
The Elder Brother of the monastery told Brienne that he knows an awful lot about the mind and soul on Sandor. Maybe too much for he only spoke to him briefly before the Hound apparently died.
The gravedigger is very large, bigger than Brienne.
The gravedigger has an injury similar to the one inflicted upon Sandor.
Brienne notices the gravedigger bend down to pet a dog. Believed to be a hint from GRRM.
The Elder Brother is very vague on the subject of Sandor's death. It is possible that he is speaking metaphorically by saying that the violent and angry Hound is no more, but Sandor still lives. This is backed up by the Brother himself as he said that he 'died' on the Trident when he was a warrior. His 'death' was a major life change, which could be the same for Sandor.
Sandor's horse Stranger was also present on the Quiet Isle, this supports the theory as it was said that only Sandor can lead or ride Stranger as the horse is exceptionally difficult to handle.
Before Arya left Sandor, she asked if he would die. His response was that he would unless there was a maester nearby. The Elder Brother is well known for his potent powers as a healer, I think this was another hint pointing towards Sandor being treated for his wounds and still lives.
However, there is one quote by the Elder Brother that could go against this theory:
"I buried him myself. I can tell you where his grave lies, if you wish. I covered him with stones to keep the carrion eaters from digging up his flesh, and set his helm atop the cairn to mark his final resting place."
Source: A Wiki of Ice and Fire: Gravedigger/Theories.
I personally do not believe we have seen the last of Sandor.
This is finally answered in Season 6 Episode 7, "The Broken Man". The episode starts with a cold open in some religious camp. At the end of the scene one of the men is revealed to be Sandor Clegane. Ray and Sandor have a conversation later on about what happened to him.
Ray: When I found you, I thought you’d been dead for days. The way you were stinking already and you had bugs all over you and bone was coming through right there. I was gonna give you a proper burial and then you coughed. Oh, nearly shit myself. I reckoned you were gonna die by the time I loaded you on the wagon, but you didn’t. Now, I reckoned you’d die a dozen more times over the next few days, but you didn’t. What kept you going?
Game of Thrones, Season 6 Episode 7, "The Broken Man"
So in short he was found on the verge of death by Ray and then nursed back to health with a few death scares along the way. Since then he's been helping out Ray at his encampment.
You can view the scene here.
His westeros.org entry summarizes the answer for the books, whatever they will do with him in Season 5 of GoT, we will be able to watch next year, presumably.
As you state in your question, that is the position he is in at the end of A Storm of Swords, what happens to him afterwards (if anything) can be found in A Feast for Crows.
It is not it. You said it yourself, he is wounded, not dead. Details in the links. Probably better to read the next book/watch the next season though.