Harry didn't bother looking for the Dursleys when he passed through the barrier that separated platform nine-and-three-quarters from the rest of Kings Cross. He headed straight for the ticket booths and bought a ticket for the tube. One that would take him to Charring Cross. He'd left the Dursleys with intention to run away before the school year had started, did they seriously think that he'd willingly go back to them now?
Not bloody likely.
Harry was a smart boy and he learned fast. When he was five, he'd discovered that he enjoyed learning. Shortly after he discovered that if he wanted to keep all of his skin intact he couldn't let his family know just how much. Libraries were his favourite places to be, and he'd ploughed his way through both fiction and non-fiction in the years he'd been living with them. It helped that he didn't have friends demanding his time actually.
Harry knew that Ron wasn't a good friend. For that matter, he knew that Hermione wasn't much better. He was actually a lot closer to Neville and the Twins than he was to the other two members of 'the Golden Trio'. He was closer to Sirius – who he'd only just met – and Remus, than he was to Hermione and Ron. Though, honestly, he was a little bit manipulative of his two nearly constant companions. He used Hermione's drive to know everything as his own excuse to Ron so that he could enjoy his reading. He used Ron to make it look to Hermione like he wasn't taking things seriously. He used both of them to give himself a level to coast at – averaging between the two of them – so that he wouldn't stand out among his peers any more than he already did.
He also used his owl, hiding his correspondence as simple devotion to his very first birthday gift, friend and treasured pet. Hedwig didn't mind though, as she actually knew and understood what Harry was doing. She carried mail spelled against being noticed, taken from her, or opened except by those that it was for between Harry, a few specific goblins at Gringotts, and occasionally an appropriate Ministry flunky – and had done since Harry had left Diagon for Hogwarts that year (previously he could have meetings in person at Gringotts bank).
As such, he knew a lot more about his situation as a powerful figure in the magical world than he suspected Dumbledore wanted him to. He even knew about the ruddy prophecy that was the reason the Dark Ponce had come after his family in the first place – and why Neville visited his parents in the long-term ward at St Mungos. A situation that Harry thought was foolish. After all, they could be cared for just as well in their own homes as in a hospital, and the familiar environment might actually help them recover.
Harry emerged from the tube at the Charring Cross station, dragging his trunk behind him, he was glad that he'd thought to put an expansion charm and a feather-weight charm on it before leaving Hogwarts that first time. Never mind that they were both supposed to be beyond him as a first year student. Harry had learned on his own for long enough that he had come to the conclusion that adults deliberately didn't teach a lot of things to children for fear of being shown up. The experience just this past year of learning the patronus charm (which was supposedly beyond many witches and wizards who had graduated already) and learning that his father, godfather, and the traitorous Pettigrew had all become animagus while at school (a process that McGonagall said took a decade at best for a fully qualified witch or wizard) was only further proof of his theory.
That, and it was easier to beat the odds when a person didn't know what they were.
Harry passed through the Leaky Cauldron and into the fire, scooping up a handful of floo powder before he stepped in.
"Potter Manor," Harry said, quietly but clearly. "In the Lake District." Then he threw down the powder and disappeared in a roar of harmless green flames.
"Master Potter is returned to us!"
Harry chuckled as he opened his eyes to see a collection of house elves standing around him. In keeping with his poor floo skills, he'd been harshly ejected and landed on his back, winded.
"Yes I have," he agreed. "And I'm expecting my godfather Sirius Black and my friend Remus Lupin to arrive tomorrow -" he'd slipped a note into Sirius' robes when he and Hermione had rescued him. He'd had time to write it after all while they were waiting for everybody to come back from the shack. "- along with a hippogriff, so please make sure that everything is ready for them."
The house elves shifted nervously.
"Master Potter would welcome escaped convict to his home?" asked one of the elves.
Harry smiled softly. "When the man is innocent and was never actually convicted in the first place, yes," he told the elf.
The elves all brightened at this news. Their master wasn't harbouring criminals, he was protecting an innocent victim. Over the summer, Harry was also planning on getting one Ms A. Bones (head of the DMLE according to the goblins) to actually provide Sirius with a trial so that the man could be cleared and freed.