He stalled for a while, planning to wait out Dumbledore's watch on him. He searched out all possible foster families in the Wizarding world, hunting for his godson before turning to the Muggle world.
First stop, Petunia Dursley. After ascertaining that she did indeed hold custody of Harry and waiting a little bit longer, he showed up on her doorstep.
He didn't really think this would work—and sincerely hoped it wouldn't, but he really wanted to raise his godson. He owed it to Harry and his parents to try.
Sirius knocked on the door and a curt Vernon Dursley answered the door. He stared for a heartbeat at Sirius' blue robes and slammed the door shut, yelling something about freaks.
Sirius Black was very nearly at the end of his tether when it came to waiting, but he bore it as he knocked again.
He heard a muffled yell of, “Go AWAY!”
Sirius sighed. “I wish to speak to you about your nephew. I would like to take him in.”
Vernon cracked the door open at that, and gestured him inside after studying him for a moment. “Can't have the neighbors think we're entertaining lunatics, so get in here!”
Sirius complied, and followed the man into the kitchen where Petunia stood. She glanced at his robe as well, and glared icily at him, “Isn't enough I have one of your kind living under my roof? Apparently not! You lot keep showing up!”
Mmm yes, he would be taking Harry away from here, come hell or high water. “I'm Harry's godfather and I would like to take him in,” Sirius said tersely.
Petunia stared at him hard, her duty to her sister forcing her to finally ask, “How do I know you're his godfather?”
“Harry has a small, circular birthmark on the inside of his left ankle,” Sirius replied promptly. For the brief time he had been able to spend with Harry so far—mostly when he had been made the boy's godfather—he had done everything from dress the baby to put him to sleep, so he knew the mark was there.
Petunia made her way out of the kitchen and Sirius turned to watch her; the woman opened a cupboard door and entered. Sleepy coos echoed out of the small room and it took every ounce of self control—of which he had precious little left—not to strangle Petunia. He was absolutely livid that a child—his godson no less!—was being kept in a cupboard under the stairs.
She returned with Harry, holding him under his arms so that his feet dangled in the air. She held him away from herself, as if she didn't want to touch him. Petunia dumped Harry into his arms. “Get out.”
Sirius cradled his cooing, wide-eyed godson, resisting the urge to scream at her. He took a deep breath, trying to reign in his temper, “I trust this will remain confidential? You will not alert Dumbledore?”
“No, we will not,” Petunia said coldly.
“If we did, he'd likely dump the brat here again,” Vernon spat. “Now, get the hell out of my house!”
Sirius jerked his head in a nod, turned on his heel, and left.
It was months before anyone heard from Sirius (Remus being the first one), and years before anybody thought to check in on Harry. By then, it was far too late and Harry had been thoroughly Marauderized.
Dumbledore managed to force Harry back to Privet Drive once when he was nine, but after Harry mocked Dudley's weight until the boy cried, Petunia and Vernon packed the boy up and drove him back to Sirius's home, refusing to take him in ever again.
“I will never have that abomination in my house again!” Vernon yelled as he flung the car door open.
Petunia cradled a fat, sobbing beach ball of child in her lap as they sat in the back seat of the car. She was murmuring to her enormous son, who sniffled pathetically and rubbed his runny nose on her blouse. She didn't seem to mind.
Dumbledore, out of options, surrendered as the car sped up the dirt road in the Cornish hill country, dirt and dust flying behind it as it careened around the snake road.
Harry looked innocent as Dumbledore turned around to look at him. “What? He sneezed on me, and his fat jiggled for seriously, like, a minute.” Harry wrinkled his nose, “I was only pointing out the truth, Professor,” he said, looking up at the Headmaster innocently from where he stood next to his godfather on the dusty path that lead to Sirius's cottage. “The enormous truth.”
That rather ruined it for the Headmaster, though Dumbledore hadn't believed the child was innocent anyway. He had been raised by the two remaining Marauders after all—if the boy was truly a naïf and not a hell raiser, Dumbledore would eat his hat.
Harry continued to look at the old man, looking for all the world like a black haired angle that had dropped out of a cathedral painting. He flashed a Sirius grin at the Headmaster when he sighed wearily. Dumbledore felt a moment of immense pity for Severus. This child was wily and charming, and would have everyone but Argus and Severus wrapped around his little finger with one wide-eyed look and an innocent smile. Severus was going to be in so much trouble when this Marauder came to Hogwarts.
Sirius hugged his godson as Dumbledore left. Oh how he looked forward to unleashing his little hellion on Snape when he got to school! That would be the stuff of legends!