Mr. Bumble is the beadle in the town where Oliver is born. As beadle, he’s responsible for running all of the "charitable" institutions in the parish—including the baby farms and the workhouse. He also gets to wear a special cocked hat, of which he is very proud. If Mr. Grimwig is the kind of one-sided character who can be reduced to an expression, Mr. Bumble can be reduced to his beadle hat. Again— super-typical of how Dickens writes minor characters.
Mr. Bumble likes power, and he likes to use it. Frankly, he’s kind of a sadist, but he’s not without a few redeeming qualities. When Oliver pours his heart out to him on the way to Mr. Sowerberry’s—
"I feel as if I had been cut here, sir, and it was all bleeding away;" and the child beat his hand upon his heart, and looked into his companion’s face with tears of real agony. (4.45)
—Mr. Bumble is actually moved. He does have a heart! But the trouble is, that he doesn’t act on his pity. He seems to feel like it’s a weakness, and he doesn’t want to lose face. He seems to think that he won’t be respected if he shows pity to anyone. This is one of just a few instances in which Dickens seems to want us to see Mr. Bumble as a real character, and not just a one-sided meanie.
Mr. Bumble’s soft spot is what allows us, later on, to feel sorry for him (but only slightly). He marries Mrs. Corney for her money (she’s not even all that wealthy), and loses his post as the beadle to become the master of the workhouse. Little does he know that the workhouse can’t have two masters, and Mrs. Corney (now Mrs. Bumble) is already it. She beats him and humiliates him, and we almost pity him—but then we’re reminded how much he used to enjoy beating and humiliating the paupers and orphans.
At the end of the novel, Mr. and Mrs. Bumble are so reduced that they end up living at the workhouse where they used to lord it over others. That’s poetic justice, right there.