I've heard that some people assume HAL starting killing in self-defense once it was revealed that he was making mistakes (the misdiagnosis of the AE-35 unit). However, I think he made the mistakes deliberately to trick the crew to leaving the Discovery One. Either way, what was HAL planning to do if Dave had not forgotten his space helmet?
I don’t think he was capable of killing the crew while in the Discover One. Redundant safety mechanisms would prevent him from say, opening both air locks and causing explosive decompression.
As an aside, I believe the suspended animation equipment contained numerous and redundant safety systems, but that HAL downloaded firmware patches into that equipment to allow him to kill the suspended astronauts. I suspect HAL did this by using forged Microsoft certificates made with a SHA-2 cryptographic hash collision. :)