Dickens introduces Uncle Pumblechook in Chapter 4, when he comes to dinner at the Gargary household. Pip, the narrator of the novel throughout, describes the man as follows:
Mr. Wopsle, the clerk at church, was to dine with us, and Mr. Hubble the wheelwright and Mrs. Hubble, and Uncle Pumblechook (Joe's uncle, but Mrs. Joe appropriated him), who was a well-to-do cornchandler in the nearest town, and drove his own chaise-cart.
This gives a picture of the social status of the Gargarys. Mrs. Joe appropriates Uncle Pumblechook because he has the most money and highest status. A cornchandler deals in grain and seed. Drivinig his horse and buggy is a symbol of status--but no doubt he needs transportation to get around the countryside in his "peppercorny and farinaceous" occupation.
It is in Chapter 7 that Pip's sister explains to Joe how Pumblechook managed to get Pip into Satis House:
"And couldn't she ask Uncle Pumblechook if he knew of a boy to go and play there? Isn't it just barely possible that Uncle Pumblechook may be a tenant of hers, and that he may sometimes--we won't say quarterly or half yearly, for that would be requiring too much of you--but sometimes--go there to pay his rent? And couldn't she then ask Uncle Pumblechook if he knew of a boy to go and play there? And couldn't Uncle Pumblechook, being always considerate and thoughtful for us--though you may not think it, Joseph," in a tone of the deepest reproach, as if he were the most callous of nephews, "then mention this boy, standing Prancing here"--which I solemnly declare I was not doing--"that I have for ever been a willing slave to?"
Dickens adroitly manages to impart a great deal of information to the reader in the form of dialogue rather than tedious prose, while at the same time characterizing the speaker as a tyrant and self-styled martyr, Pip as an abused child, Joe as a long-suffering husband, and Pumblechook as a man of higher social status than the Gargarys but of no importance to an aristocrat like Miss Havisham.
The eccentric Miss Havisham did not bother to explain to Pumblechook, who was standing in the hall talking to her through her closed door, why she should want to have a boy come and play at her house, and he was probably afraid to ask. He is just grateful to be able, at no cost to himself, to be able to ingratiate himself with Miss Havisham and with the Gargarys at the same time. No doubt he hopes to benefit from having a family member gain access to this upper-class home--although it turns out in subsequent chapters that he has never even met the owner and probably never will. When the family is cross-examining Pip about his visit, Pumblechook asks:
"Boy! What like is Miss Havisham?"
"Very tall and dark," I told him.
"Is she, uncle?" asked my sister.
Mr. Pumblechook winked assent; from which I at once inferred that he had never seen Miss Havisham, for she was nothing of the kind.
Miss Havisham's motive for wanting a boy to come and play at her house has to be inferred from the way Pip is treated when he goes there. Evidently the old lady is lonely and wants company. At the same time she would like to have a vulnerable young male for her niece Estella to begin to play cat-and-mouse with in order to have practice for tormenting the male sex in the future, as well as to provide some entertainment for her vengeful foster mother.