The most crucial moment when he actually snapped.... (***Spoilers***)
….was when he saw the photo of his mother which had 'Love you' written on the back side by Thomas Wayne with his initials.
It was at this point, he realised that his mother was right about the whole relationship. He was almost devastated so much that it was beyond mourning for him. He was terribly speechless to realize that he murdered his innocent mother.
An ultimate jolt of shock it was. He turned truly psychotic & hopeless at this point and its here he got transformed into 'Joker' in true sense.
The whole flood of anger, frustration, agony was unleashed on Randall. You can see his wet eyes when he sat down after killing Randall.
Before this crucial snap, he always had an intention to commit suicide at Murray's show.
But its after this event, he decided not to but kill Murray.