tvtropes calls it Midnight on the Firing Line, which is kind of ironic, that something that must have occurred to literary craftsmen throughout the ages should be baned after a mid-1990s Television show. Whatever it is called, the process of setting up the premise and conflicts of an episodic drama through a first episode that at first seems somewhat between prosaic and mediocre is something that is hard to capture. Especially when that first episode involved extra helpings of dramatic irony that are only revealed after the fifth reading. This is especially sharp because like many Harry Potter readers, especially the ones who appreciate the overall arc of the works, I usually analyze the first book to be merely an introduction to the series, and the first chapter of the first book to be a paragon of accessibility that can be passed over in haste. Not so! The first chapter has so many signs and portents of the entire series that it only makes sense after reading the entire series, and the entire series only makes sense after reading it.
Mr. and Mrs. Dursley... The chapter and series begins with these words, and these were the first characters introduced. Them being introduced serves several purposes. First, it allows the reader to move from the common world of muggles into the wizarding world. Second, while the first scene, first chapter and much of the first book deals, in a comic fashion with the Dursley's intolerance of the magical world, the series will reverse that soon enough, dealing with the magical world's intolerance of muggles. Third, one of the major themes of the book is not the battle between good and evil, but the battle between good and bad. The book opens not with either the best nor the worst, but with the most mediocre people imaginable. The Dursley's are hyperbolically mediocre, much as Cornelius Fudge and Dolores Umbridge will later be.
...as though it had been broken at least twice. From the introduction of Albus Dumbledore. A terrific example of conservation of detail, since Albus' broken nose is finally explained in the final book, and is actually quite significant. I have a theory that the other times it is broken might explain one of the few unexplained mysteries of the series, as well.
"My dear Professor, I've never seen a cat sit so stiffly." This line towards Minerva McGonagal explains both some of the seriousness of McGonagal, and Dumbledore's own seeming levity. This is continued throughout their dialog, where Dumbledore shows his foregiving nature towards the ongoing celebrations, remarking "We've had precious little to celebrate this past 11 years"
"Voldemort had powers I will never have"
"Only because you are too --well-- noble to use them."
This line of dialog between Dumbledore and McGonagal seems odd, in that throughout the books, Dumbledore is shown as a paragon of goodness. Again, in the last book, the fact that Dumbledore was at one point involved in dark magic, and could have gone that route, becomes an important plot point.
"...how in the name of heaven did Harry survive?" Although I do not know for certain, I think the mention of heaven in this question is not merely a verbal expression: I think that the question contains the answer, and is a hint that Harry survived through a larger moral providence.
"I would trust Hagrid with my life." This line struck me as perhaps a very, very early hint of dramatic irony. He trusts Hagrid with his life, but only trusts Severus Snape with his death.
"Scars can come in handy. I have one above my left knee that is a perfect map of the London Underground" The exact use of the scar, which Dumbledore may or may not suspect, becomes more and more apparent as the series progresses. His mention of the London Underground, although probably a lie, shows that he is much more accustomed to the muggle world than many of his fellow wizards.
Then, suddenly, Hagrid let out a howl like a wounded dog. Severus Snape is also described as howling like a wounded animal, when his reaction to the death of the Potter family is shown in flashback in the final book. Is this meant to show parallels between Hagrid and Snape that would not otherwise be obvious?
It might be noted that I didn't cover the entire chapter, and covered single lines in greater detail than might be obviously needed. That is because every time I go back and read this first chapter, new and different aspects of it are shown to me. Even the first chapter has many secrets still to be shown. Some of them may be significant, some may not, and it is the reader's process to decide which is which.