Lady Catherine de Bourgh is an aristocrat who thinks quite highly of herself. She believes that she has all the knowledge of art, music, education, and society that could be had, and even extends her self-glorification to the point of becoming a patroness to artists, clerics, and musicians such as Mr. Collins.
She is comical because she is so into herself, and because her ego is so inflated that makes her do things and say things in ways that are quite exaggerated. First, she meddled into everyone's lives telling them what they are doing right or wrong. Second, she has totally taken over Collins, whose extreme admiration of her rank is conversely a feed for her to get even greater airs. She decided for the decoration of his house, approved his marriage to Charlotte, consistently tells him what to do and how, and she does the same with Charlotte as well.
She is also comical in her mannerisms, and how Austen portrays her in her overly-posh pose with her chin always as high as her ego. It is the little things she does, and the ways she says things, plus her sense of self importance that, put together, make a huge mosaic of the eccentric and haughty personality of this character.
I would add that the comical aspects of Lady Catherine de Bourgh are not limited to her absurdly proud, vain character. The woman's circumstances and her intereaction with the other characters--especially Elizabeth--provides some amusement as well.
First Lady Catherine's pride in her rank makes her somewhat silly when we observe that she lacks the courtesy one would expect to find in one of her class.
Second, Lady Catherine is determined to marry her daughter, Anne, to Darcy. The fact that Anne is "sickly and cross" and not even able to go into society is ironic, and the circumstance is humorous (although we of course most readers will feel sympathy for Anne).
More importantly, Elizabeth, unlike most of the others, refuses to be intimidated by Lady Catherine. Unlike her friends, Elizabeth fails to give Lady Catherine the fawning attention she expects.
Elizabeth's final triumph comes in Volume III, when she uses particularly deft, assertive speeches to thwart the woman's plans for Anne and Darcy and defend Darcy's right to marry whom he chooses (See Chapter 24). In this final scene, Lady Catherine's fury and impotence make her a pathetically comic figure.