He doesn't really know what it is.
He remembers being a child with an extreme lust for battle- the same lust he has now. So the differentiation between what he is now and what he was then means little to him. Power is the difference.
Then he meets a girl without a name.
She has an aura of power- but he knows it's nothing compared to Unohana Yachiru's strength. All he used to do in the face of someone with any form of reiatsu was challenge them. Not now. There was blood in front of her and a look in her eye he never understood.
"Are you going to try and kill me too, mister?" she asked with an unmistakably calm voice. Her eyes showed empty fear. Of course- this was a place where fear was so common that it meant nothing. Everyday was a quest to die for most people here.
He did not kill her. He can't grasp what it is about her that makes him not want to kill her. His goal is to seek out fights, and this small girl is not something he would gain much from fighting. There is power. . . but not the power of the one he is seeking.
So he bends down and puts his hand on her head like it's nothing.
"Shut up ya little brat," he snarls. There is something new in her eye.
"You're that guy who beat up that Captain aren't ya?" she said with a gleaming smile. His eyes widened and shock filled him.
Her eyes weren't filled with fear or empty loathing. She was filled with admiration. Somehow in this hell of a place there was a little girl unlike any other. Though she was able to fight, her eyes weren't filled with blood lust. Though she was in a hell like place set out to kill her, she was not afraid to hope.
So when she answered his next question, what is your name, with not having one- he knew then.
"Come on, Yachiru," he said dryly as he walked away. To his surprise she didn't timidly follow. She clung to his shoulder and he could hear her laughter.
"You don't even have to ask, I'm always gonna follow Kenny!" He grinned madly.
"That's a bold promise for a kid like you who just met me!"
But it was true, she always followed him from then on.
And in the dark hell of District #80 Kenpachi Zaraki held out his hand and found what it was to be an 'adult' through the eyes of a child filled with promise.