Short answer: Alliser Thorne is a very small man, and Jon stepped on his toes.

Long answer: there are many factors that helped shape the relationship between Jon and Thorne. For starters, Alliser Thorne was a Targaryen loyalist  who was sent to the Wall on the command of Tywin Lannister after the Sack of King’s Landing at a time when lots of loyalists were getting pardoned after bending the knee, something that clearly embittered him and left him resentful of the Rebellion and the Night’s Watch itself. Meaning that he was predisposed to disliking Jon, being that he was the son of one of the rebel leaders. Not that Jon would have been spared if he hadn’t been Ned Stark’s son; from what we’ve seen of Thorne’s behavior, he was predisposed to disliking all the recruits with equal fervor. Because Alliser Thorne is a bully who preyed on those weaker than him and who took out his resentment on his recruits. He always found a reason or another to hate his recruits which he took as probable cause to demean them.

The bigger shift from being one of many targets of Thorne’s indiscriminatory dislike of all recruits to being the target of his impassioned hate happened after Jon got the news that Bran woke up. Jon – who had previously received a verbal lashing from Donal Noye that smacked the castle-bred superiority right out of him and who was in a very cheerful mood after he got the news that Bran was going to live – offered to help Grenn with his training, essentially offering to do Thorne’s job for him, something that Alliser took ill to.

Alliser Thorne overheard him. “Lord Snow wants to take my place now.”
He sneered. “I’d have an easier time teaching a wolf to juggle than you
will training this aurochs.” 

“I’ll take that wager, Ser Alliser,” Jon said. “I’d love to see Ghost juggle.”

 Jon heard Grenn suck in his breath, shocked. Silence fell.
 Then Tyrion Lannister guffawed. Three of the black brothers joined in
from a nearby table. The laughter spread up and down the benches, until
even the cooks joined in. The birds stirred in the rafters, and finally
even Grenn began to chuckle.

Ser Alliser never took his eyes
from Jon. As the laughter rolled around him, his face darkened, and his
sword hand curled into a fist. “That was a grievous error, Lord Snow,”
he said at last in the acid tones of an enemy. 

As far as Thorne was concerned, Jon publicly mocked and undermined him in front of the whole Watch, most of them his subordinates. For a prickly man who seems to have the same relationship with laughter as Tywin Lannister, and who found pleasure in holding his superiority over those weaker than him, that moment was enough to earn Jon his everlasting enmity. Jon effectively broke Thorne’s power by his cheeky comeback, something that no self-respecting bully would take kindly to. Jon was on Thorne’s shitlist from this moment onward.

It was a slippery slope from then on out. Because Jon continued to challenge and undermine Thorne. He showed Thorne up making it very clear to everyone in Castle Black how incompetent he was at his job. Oh he was just abysmal; his “training” consisted of yelling at the recruits and making them beat each other bloody without moving a muscle to actually teach them anything. He even denied them their personhood by coming up with nicknames that were either build on their physical appearance (and that relied on comparing them to animals) or were designed to hit them where it hurt (Lover for Daeron who was falsely accused of rape. Lord Snow for Jon whose bastardy was an open and easily recognized wound, etc). Thorne was content to make the recruits feel useless and denying them the means to prove him wrong. Until Jon stepped in to work with them and impart his own training on them, proving just how wrong Thorne was about them. The “Aurochs” made significant progress to the point of being named a ranger, as did Pyp who sought Jon out to learn from him because Thorne never bothered to show him how to grip a sword properly. Jon, more or less, replaced Thorne as a trainer and showed everyone that the problem was less about the ability of the recruits and more about how they were trained and who trained them.

So not only did Jon prove Thorne incompetent, he proved him willfully incorrect in his assessment of the recruits. Basically, an idiot who had no idea what he was talking about. He stood up to Thorne repeatedly as well, from refusing to give up on his fellow recruits as a lost cause to stepping in to protect Sam in the yard, impressing the shame of still going after Sam after he yielded on his fellow recruits. Which publicly shamed Thorne as well considering that he was an anointed knight clearly breaking his vows and every chivalric code. Jon undermined Thorne again when he managed to convince the other recruits (or in Rast’s case, threaten) not to beat Sam up on Thorne’s command, and again when he went to Maester Aemon for help to get Sam out from under Thorne’s thumb.

And through it all, Jon was enabled in his defiance of Thorne by the highest ranking officers on the Wall: Lord Commander Mormont and the widely respected Maester Aemon, which is a reflection of what they thought of both Jon and Thorne.
Maester Aemon listened to Jon and went over Thorne’s head to get Sam.

Jeor Mormont denied Thorne the sick satisfaction he got from Jon’s appointment to the stewards by appointing Jon as his personal steward, a position understood to be given to someone being groomed for command. And it wasn’t only them who showed Jon favor.
Tyrion – someone Thorne actively loathed for being both a Lannister and
dwarf – laughed off Thorne’s humorlessness and got the rest of the high
officers laughing at him, but he lent Jon his aid which consisted of
twisting Thorne’s arm right in front of Jon by forcing him to tell Jon
that Mormon’s summon had to do with Bran. The recruits defied Thorne and
complied with Jon’s demands instead. Even when Thorne’s cruel barb about Ned succeeded in provoking
Jon into attacking him, Jon promptly distinguished himself saving
Mormont’s life the very same night and his punishment boiled down to
being held in his room for a while.

With Jon established as Thorne’s enemy (which honestly says so much about the guy), it must have stuck in his craw that Jon was being shown favor from the same people who mocked Thorne himself, and was allowed and even rewarded for his defiance.