In the Deep Space Nine episode "Tacking Into the Wind", Worf challenges Gowron, the Klingon leader to a duel, defeats him, and becomes the "leader of the Empire" (which he later passes on Martok). If defeating the Chancellor in a duel allows the winner to become Klingon leader, why wasn't the Klingon leader constantly being challenged by numerous other Klingons? Say by many members of the Duras family or others? Why, for example. didn't Duras challenge K'mpek in TNG? He was old and weak and surely no match for Duras?
The scene involves Worf, because the other Klingons at the table wouldn't say it, call out Gowron for his bad leadership. Gowron does not take it well.
Gowron: If you were a true Klingon, I'd kill you where you stand! Fortunately for you, that child's uniform shields you from your rightful fate.
Worf: (takes off commbadge) What I say now I say as a member of the House of Martok, not as a Starfleet officer. You have dishonored yourself and the Empire, and you are not worthy to lead the Council.
Gowron: There can be only one answer to that! (Throws Chancellor's cloak on the table. He and Worf go pick up weapons.)
Gowron is a jerkass so there'd be only one way he'd likely respond, which Worf was obviously anticipating because he was completely calm and unsurprised. The way the scene played out made it clear that the only reason Worf felt comfortable doing it was because he knew he had the support of the Klingon leadership in the meeting, even if they wouldn't speak up directly. And that's the key; he had to believe they would support him, even silently.
The whole thing actually started when Worf had the discussion with Ezri where she pointed out Gowron's bad leadership and the complacency in the Empire that allowed leaders like him. The implication is that Chancellors (and probably House leaders) wouldn't be overthrown by challenges on a regular basis because they would really have to tick off enough important people that a challenger would ever get to the point of being able to get a duel to begin with because most of the important people were quite happy with the status quo. They'd accept whatever method would be used by the challenged leader to dismiss the challenger, if they didn't step in themselves to deal with the upstart one way or another. A crewman who challenged a ship's captain, for instance, without the support of the crew or a good chunk thereof would probably announce their challenge and immediately be shivved in the back with a d'k tahg by another crewmember who promptly states they just stopped a dishonorable mutiny, the captain commends them for their loyalty and diligence, and the would-be challenger is dumped out an airlock.
This would tend to make would-be challengers a bit more circumspect.
Aside from that, note that the lead-up to the Klingon civil war shows that challenges aren't simply open to whoever wants to make one for whatever reason. When Toral challenged Gowron, Picard dismissed the challenge, as per Klingon tradition and law, on the basis that Toral hadn't proven himself as a warrior yet. So there are ways of avoiding duels that are not seen as dishonorable. Logically, you would expect things like that to be present in the culture as a safety measure to prevent, for instance, an otherwise good leader who'd made a relatively minor honest mistake from being killed by some ambitious underling who would turn out to be incompetent in the role.
Normally, Gowron probably would have laughed off the idea of being challenged, and had Worf booted out of the room on some pretext--an obvious one is that he did not want to create a problem with Starfleet, which Worf was still a member of. However, leading up to this it was made clear Gowron was becoming more unpopular among the warriors and other Klingon leaders, had demonstrated significant strategic and tactical missteps, and was paranoid of the popular and successful Martok.
So this was a case where Worf knew that Gowron was losing support and, more importantly, Gowron knew he was losing it. When Worf stated he was saying something the others wouldn't, no one argued or rushed to defend Gowron, or denounce Worf for his accusations, or tried to distance themselves in any manner. Gowron was already being considered (and had been called) a politician and not a warrior. In other situations he might have gotten out of it, but at that moment he couldn't because he'd appear weak, and that would allow someone to move against him openly, such as Martok, and any support Gowron had left would be gone due to his perceived cowardice.
The Klingon Bird of Prey Owners' Workshop Manual gives a pretty decent overview of how and why a Klingon can reasonably challenge their superior. In short, a challenge can only be offered under certain circumstance (when the incumbent shows weakness, for example) and can't be offered by someone more than one step below the office holder unless they've offered that person a personal insult.
Officers are prevented from abusing their power because each crewman literally has the duty to challenge his superior if he sees any signs of weakness, cowardice, failure to perform their duty or dishonorable behavior. Each member of the crew can only challenge his direct superior and there are strict rules for how a challenge can be made and under what circumstances. If a challenge is successful and the crew member kills his superior, he takes his place and advances in rank, but challenges are not undertaken lightly, if a warrior succeeds in killing his former superior, but does not have the support of the crew, it is inevitable that he himself will be challenged and this would almost certainly lead to his death.
There are also limits to how far a Klingon can advance by challenging his superior. A servant cannot become a warrior as the result of a challenge, and a warrior cannot become an officer.
So who can challenge K'mpec and when can they challenge him? In short, the challenge needs to be a) seen to be honourable and b) offered by the head of another of the Great Houses which means that there are tight limits on the circumstances that a challenge can be offered and the number of potential challengers.
Since K'mpec has the open support of all of the other Great Houses of the council, no legitimate challenge would be accepted. It would be seen as dishonourable to challenge him simply because he's old (unless he was actually seen to be infirm or incompetent as a result of his age) and any challenger would either be executed on the spot for their dishonourable conduct or, if they could somehow provoke him into accepting the challenge and then kill him, be immediately re-challenged and have to fight a succession of match-fresh opponent/s until they themselves were defeated.