I've just watched episode 20 of Hunter X Hunter, in which it is revealed that:
During the final phase of the Hunter Exam, Killua kills Bodoro as soon as the fight between Bodoro and Leorio starts.
Why would Killua act in this way?
He knows that both interfering in other fights and killing other applicants is against the rules.
So I don't understand why he did what he did.
There are a few factors in play, but the most prevalent reason is Illumi's control over Killua. It's possible you underestimate just how much of an influence Illumi exerts—shown both in this episode, and elaborated on in episode 94.
Suffice it to say, Illumi basically raised Killua to be a killer, and thus has had years and years of exposure to Killua to manipulate how he thinks and behaves. Through his aura and manipulative skills (and other methods, revealed in episode 94), he was essentially able to convince Killua that he was not worthy of love or friendship, and that all he was meant to be was a perfectly honed killing machine.
Essentially, this reversal of psychology drove Killua into an unstable state, unearthing his urge to kill and forcing him to act upon it again. At that point, he had been driven to believe he was just a killer, had failed to protect Gon, and he'd never be a Hunter—why would he care what happened anymore?
(As a side note, it also ensured both that he wouldn't have to fight Leorio, and that Leorio would become a Hunter. So bonus points to Killua for that.)
He did it because it was against the rules. He became disqualified on purpose.
Killua has been influenced by his family since the moment he was born, being taught to believe that he could do nothing other than kill. If Illumi told him he wasn't worthy of a friend and shouldn't be a hunter just yet, Killua would purposely fail the exam.
Illumi had basically told him he would end up wanting to kill Gon if he stayed with him.
In the movie, Phantom Rogue, Killua can't even move when faced with the prospect of fighting even a doll of his brother. It just shows how much influence Illumi has over Killua.
It's been shown multiple times that during the earlier years of hxh, Killua had little to no hesitation about killing and, although it's said he's sick of killing, I think he stopped killing later on more because Gon said so, rather than he thought it was morally wrong. Therefore, exiting the final exam in such a dramatic way is not so shocking by his standards.
Killua's got a huge amount of bloodlust underneath the exterior, as it becomes apparent when he goes into 'assassination mode'. Killing would be a way of venting anger and frustration.
I guess a part of Killua would want to cut off all chances of becoming a hunter as soon as possible because that is what his brother would have wanted. If he continued to fight in the tournament, a person might randomly forfeit to let him win. Unlikely, but still possible.
And finally, a tinsy, winsy part of him might have wanted to help Leorio.
