Harry's Aunt Petunia has no magical ability and she thinks her nephew is a freak as a result. She and her husband have "raised" Harry since he was a baby and housed him every summer vacation since he started Hogwarts, but they both hate Harry and wish he weren't related to them.
Most of Aunt Petunia's behavior in Book 5 is in keeping with her usual bigoted, emotional abusive nonsense. She keeps on treating Harry like a shameful secret the Dursleys have to house against their will. But there is one intriguing breakthrough in her character that really starts to make Aunt Petunia seem more human and less cartoonishly cruel and stupid. After her son has nearly been killed by dementors, Harry tells his furious aunt and uncle that Voldemort is back. And his Aunt Petunia, whose sister Lily was Harry's mother, actually seems to know something about what that means:
She was looking a Harry as though she had never looked at him before. And all of a sudden, for the very first time in his life, Harry fully appreciated that Aunt Petunia was his mother's sister. He could not have said why this hit him so very powerfully at this moment. All he knew was that he was not the only person in the room who had an inkling of what Lord Voldemort being back might mean. [...] The furious pretense that Aunt Petunia had maintained all of Harry's life – that there was no magic and no world other than the world she inhabited with Uncle Vernon – seemed to have fallen away. (2.194)
This moment of surprising mutual understanding between Harry and Aunt Petunia, in which he suddenly remembers that his mother who was murdered by Voldemort was also Aunt Petunia's sister, reminds us that the worlds Harry inhabits aren't so separate after all. From Book 1 onward, the wizarding world and the Muggle world look completely different. But beginning with Book 5, we start to remember that the Muggle world is also filled with Squibs and Muggleborn wizards. As Voldemort rises, Muggles are no safer than magical people, and Aunt Petunia seems to realize that the boundary between the magic world and her world is starting to break down.
Of course, having Harry in her home is a threat to the safety of her whole family. In Chapter 2, Uncle Vernon tries to throw Harry out of the house so that Voldemort doesn't come and find them. But Aunt Petunia receives a Howler – a special enchanted letter – that says only, "Remember my last, Petunia" (2.217). By the final chapters of Book 5, Harry discovers that this Howler was sent by Professor Dumbledore to remind Aunt Petunia of her oath to give Harry a place to live. Aunt Petunia's relation to Lily, who was killed by Voldemort out of love for Harry, provides a special, ancient magical protection for Harry. Voldemort cannot find Harry as long as he has a home with Lily's blood relations. So even if the Dursleys don't accept Harry into their home willingly, they still accept him. This magic is what keeps Harry safe for the early years of his life.