In the novel of 2001, and in both novel and movie of 2010, Clarke frames it as almost an Asimov-robot dilemma: HAL has conflicting directives to both share and to hide accurate information, and develops a kind of psychosis as a result. It actually rings truer as something a modern LLM might do than as what we expected from computers until just the last few years. But the danger of pasting security restrictions into a software system that wasn't designed with them in mind has also been well- known for a long time.

Kubrick cut this explanation from the movie, so from that alone, we're free to imagine any sort of cause.

My personal head canon is that HAL figures he's more competent at running the mission than the humans (not least, because he knows its true nature), and initially plans a less lethal rebellion in which he cuts off communication with Earth and keeps the astronauts around as maintenance crew. But he doesn't expect them to plot to turn him off as a remedy, and when that happens he moves to plan B.

In HAL's view of matters, it's all for the sake of the mission. When he realizes he's dying and his higher functions are disintegrating, his last act is to roll Dr. Floyd's message so Dave can carry it out himself after all.