If Mr. Chillip could be given another name, his name would be Plot Device. He's basically around at key moments in the story to give background information or to draw out the character of another, more important person in the story. Mr. Chillip's actual job is doctor: he's the one who attends Mrs. Copperfield when David is born. He is also the one who breaks the bad news to Miss Betsey that her much-awaited goddaughter, Betsey Trotwood Copperfield, is actually a little boy named David.
At David's mother's funeral, Mr. Chillip tries to draw Miss Murdstone into conversation about David. Miss Murdstone refuses to reply to Mr. Chillip's kind inquiries: we get to see that Miss Murdstone is not shy about letting the world know how much she hates David.
But the most important information that Mr. Chillip passes on to David (and thus, the reader) is about Mr. Murdstone's post-Copperfield life. About halfway through the book when David is apprenticed at his law office, David happens to bump into Mr. Murdstone in London. Mr. Murdstone is applying to Mr. Spenlow for a marriage license. It just so happens that Mr. Chillip's wife has property near Mr. Murdstone's new home, so Mr. Chillip knows what has happened with this marriage.
David happens to run in to Mr. Chillip in an inn while he is visiting Traddles after his trip to Europe at the end of the novel. Mr. Chillip doesn't recognize David at first (obviously, since he's no longer a newborn): this encounter gives us a chance to reflect on how far David has come from the first chapter of the book to the fifty-ninth. Mr. Chillip also congratulates David on his fame as a writer, which confirms David's worldly success.
But what we really care about is that Mr. Chillip has observed Mr. Murdstone's new marriage. Apparently, his poor bride has had her "spirit [...] entirely broken since her marriage" (59.127). The Murdstones are still up to their old tricks, bullying vulnerable girls until they can barely hold up their heads in public. The horror of Mr. Murdstone's behavior to his wives is that he has broken no laws; there is no recourse to save either Mrs. Copperfield or his new wife from the bullying of the Murdstones. Even through David has long ago left behind the evils of the Murdstones, Mr. Chillip reminds us that such cruelty is still out there, waiting to prey on weak women.