So Tom Riddle was described as sort of "handsome" and stuff in the second book.
Then he went someplace and completely changed his appearance, and became snakelike and stuff. I believe he is in the middle of the transformation in the middle when he asks Dumbledore for the job.
How does he do it?
Voldemort split his soul, and put portions into various objects. So long as they were intact, Voldemort himself could not die, though he could be reduced to a barely living state.
The process of creating Horcruxes has dehumanized him both mentally and physically. As Dumbledore said:
"I believe that Voldemort is now so immersed in evil, and these crucial parts of himself have been detached for so long, he does not feel as we do."
—Harry Potter and the Half-Blood Prince
With each Horcrux that Voldemort made, his appearance became more and more inhuman. When Voldemort applied for the post of Defense Against the Dark Arts Teacher, his face was markedly changed, as shown in Dumbledore's memory:
Harry let out a hastily stifled gasp. Voldemort had entered the room. His features were not those Harry had seen emerge from the great stone cauldron almost two years ago: They were not as snake-like, the eyes were not yet scarlet, the face not yet masklike, and yet he was no longer handsome Tom Riddle. It was as though his features had been burned and blurred; they were waxy and oddly distorted, and the whites of the eyes now had a permanently bloody look, though the pupils were not yet the slits that Harry knew they would become. He was wearing a long black cloak, and his face was as pale as the snow glistening on his shoulders.
—Harry Potter and the Half-Blood Prince
At the point at which this scene took place, Voldemort had not created all his Horcruxes (Nagini and Harry were still left, at the very least), and so he did not look as monstrous as he did later. He probably had four or five Horcruxes by this point.
Similarly, when Tom Riddle visited Hepzibah Smith, he had a "red gleam in his [...] eyes."
Pick it up, have a good look!" whispered Hepzibah, and Voldemort stretched out a long-fingered hand and lifted the cup by one handle out of its snug silken wrappings. Harry thought he saw a red gleam in his dark eyes.
—Harry Potter and the Half-Blood Prince
At this point, he had not turned Hufflepuff's cup into a Horcrux, nor likely Ravenclaw's diadem. He most likely had made the ring and diary into Horcruxes, and so had two.
So each Horcrux renders his appearance paler, more snakelike, less human.
Voldemort's outward appearance became more snake-like as he experimented more and more with Dark Magic and tearing his soul apart repeatedly. Every time he murders someone, he becomes less a human and more a monster*, if you will.
To quote Slytherincess's answer on a related question:
There is a scene where Dumbledore shows Harry a memory of Voldemort after he resurfaced after being gone for ten years. He came to hide the Ravenclaw Horcrux and to ask Dumbledore, who was newly appointed headmaster, for the Defence Against the Dark Arts position, and his appearance is drastically changed ... from when he had gone to work at Borgin and Burkes ten years prior. ... Voldemort was deeply involved in Dark Magic
As to exactly how he changes, the books don't specify; it's a general combinations of Dark Magic related deeds including murder, creating horcruxes, practicing Dark Magic, and doing other evil things. His physical transformation is a result of his spiraling descent into the unnatural horrors that Dark Magic brings.
* You could say he was "more monster now than man; twisted and evil" ;)
