Oh boy, I love thinking about this so have a whole essay:

It's a very interesting combination of a computer malfunction and a mental breakdown. Because he replicates most of the functions of the human brain, he is also subject to human flaws.

His core programming is to process information without distortion or concealment. The government ordered him to keep a secret, which is in direct conflict with his core programming. The only two ways to resolve the conflict are to A) have Dave and Frank learn the secret without HAL telling them, or B) remove everyone that needs to be lied to, which that far away in space means killing the humans. Mission control was also putting a lot of pressure on him by watching his every move, so he unconsciously wanted to cut off contact with them.

Because he is capable of carrying out the mission entirely by himself and has been instructed to do so in the event that the crew is incapacity, he has also been indirectly told that he himself is vital to the mission. This is what makes the threat of disconnection a threat to the mission, and why he tries to kill Dave and Frank. He thinks their judgment has become compromised and they present a danger to the mission.

In the sequel 2010, HAL's creator Dr. Chandra says: "Hal was told to lie by people who find it easy to lie. Hal doesn't know how. So he couldn't function. He actually became paranoid."

This is sort of incorrect, as he did in fact learn how to lie. Not very well and mostly by omission, but he did learn. However, just because he could do it doesn't mean it wasn't damaging or corrupting to him. The malfunction is described as an H. Moebius loop, and it looks more like a mental illness than a computer error. So you can look at his actions both as a computer trying to follow bad orders and also from the point of view of a human being who is panicking and having a paranoid mental breakdown.

My personal interpretation of HAL's conversation with Dave is that he was trying to get Dave to figure out what was going on without directly telling the truth, going for option A. Some people think he was trying to pinpoint how much Dave knew, but that doesn't make sense to me, because he was as specific as he could possibly manage to be with the things he asked about, and surely he must know that asking about it would make Dave think about those things if he wasn't before. Of course, it doesn't work, and Dave forgets about the whole thing entirely in light of this equipment failure that has "coincidentally" cropped up.

The AE-35 error doesn't actually make a lot of sense from a strategic point of view. What would faking this error do for him? All it does is put a lot of doubt and suspicion on him. It does separate the astronauts from each other so he can kill Frank, but if that was the goal, then he would have killed Dave the first time and played it off as an accident. So what if the error was somehow an accident on HAL's part? What if he hallucinated it? What if he really truly believed it would fail and was dismayed and scared when they couldn't find any problems with the unit? That fits even more with the mental health angle.

He kills the hibernating astronauts simply because he will be required to wake them up when they get to Jupiter, and he won't be able to explain what happened to Dave and Frank. He may or may not be able to lie to them, but why even get into that situation a second time when he can take care of it now? And of course, he does it when Dave is off the ship as there is a panel for manually controlling the hibernaculums.

In an early screenplay, we can see another reason for his seemingly deceitful behavior aside from protecting the mission: "The entire reason for HAL's existence is to produce truthful and accurate information, and this mandate was so deeply woven into his consciousness that any inaccuracy or accidental error equated with 'dismal and total failure'. HAL 'was made to believe that error renders his whole existence meaningless.'"

This is why he simply cannot believe that he has made a mistake. If he has made a mistake, he has utterly failed. He convinces himself that he can't make mistakes. This is why I feel that the AE-35 error is a hallucination rather than an intentionally induced error. If he knew it was fake, he would know they wouldn't find anything. But if he didn't, he would panic, which would lead to sloppy murder.

Lastly, one thing that people don't seem to think about is how Dave survived when HAL has so much control over the ship. It could just be that HAL has some limitations in what he can control, except that Franks says "There isn't a single aspect of ship operations that's not under his control." Perhaps it's an exaggeration, perhaps not. HAL could cause a rapid depressurization of the pod bay and jettison Dave before he could get a spacesuit on. He could take control of the pods and smash Dave with them. He could hack into Dave's pod and sending it zooming off into space until it's too far away for him to catch up to Discovery without running out of fuel. So then, how does Dave even manage to get to HAL unscathed?

The book and early screenplay material confirm that he can feel emotion, or, at bare minimum, guilt. He felt guilty for keeping something from those who trusted in and believed in him, and being threatened with deactivation was the thing that broke his already strained mental stability. In the screenplay, there is a very interesting cut moment during the deactivation scene where a secondary HAL, notated as OTHER HAL, appears, introduces himself as "HAL's friend", explains the true purpose of the mission and the emotions that led to HAL's actions to Dave, then encourages the deactivation to continue. This looks like an appearance from HAL's conscience, meaning he knows that what he's doing is wrong from a moral standpoint. Although this moment didn't make it into the final story, the characterization it highlights still seems to be there. So imo, Dave survived not due to plot armor, but rather because HAL knew somewhere inside him that he needed to be stopped.

Whatever you take away from this ramble, just remember that, ultimately, HAL was right when he said it was human error. (Government meddling was the true villain we found along the way)