Bounderby was falsely humble, for one thing. He was always proclaiming, and actually bragging about, his humble roots. Basically, while he pretends humility, he was completely distorting the meaning of humility.
When Dickens referred to him as the "Bully of humility," the picture I got was of a man who was browbeating or pushing around and disdaining the truly humble while hypocritically claiming to be one of them. He was always demanding homage and respect, while stomping on those he considered beneath him as though he had a moral superiority to them. In other words, he was shown to be a hypocrite, not a humble man.
I'm not explaining this well. The term, "bully of humility," is a clever paradox which shows you the very nature of Bounderby. He's a contradiction: always telling everyone he was humble, but, by saying this all the time, he was showing he was not humble at all. It was pride and conceit, not humility. (Did I just make this as clear as mud?)