Kilgrave was born Kevin Thompson. Kevin had a serious and terminal neurodegenerative disease. His parents subjected him to various painful and experimental procedures to attempt to cure him. Other children were kept and experimented on by them and other scientists at the college, though their ultimate fates are currently unknown. Kevin ultimately appeared to be cured, though the experiments had a (possibly?) unintended side effect of imbuing him with his powers. He pleaded for his parents to stop and they bent to his will.
What followed was a time of Kevin stretching his powers. He controlled his parents, forcing them to attend to his every whim and lashing out with violent tantrums when displeased. He made his mother burn her face with an iron, and when the chance came they abandoned their son, fleeing to protect themselves, constantly on the move.
Kevin was left on his own, forced to use his powers to get by, ordering others to care for him, amuse him, and growing into the man who would become Kilgrave.
He fashioned the name for himself and made great strides to erase all traces of Kevin Thompson from history. He had anything, and everything he could ever want at a whim, leaving a trail of broken toys where ever he went. People didn't hold his interest long.
And then there was Jessica.
He happened upon her in an act of heroism, and was drawn to her. She was like him, and that made him want her. He told her to come with him, and the time that followed was- to him- wonderful. In his mind he fashioned an epic love story for the two of them; a whirl wind romance. He was obsessed with the idea of her. After a period of months he even tested the waters, carefully letting the clock run out on his orders. It was a brief moment of freedom, but she stayed, she was his.
In actual fact, even apart from the mind control forcing her every action Kilgrave was in almost every facet an abuser. He distanced her from her friends and family, forced her to tell lies to keep them from worrying, and molded her to fit his desires. He manipulated her every move, and lashed out violently when things didn't go his way. He nearly had her slice off her own ears, stopping her at the last moment as though it were an act of mercy, a show of his affection and love for her. He was almost completely oblivious to her desire to kill herself in that moment of freedom, and he freed himself of any possible internal doubt that she didn't want to be with him.
But the real obsession, that came later.
Reva Connors came into possession of a tape showing evidence of the experiments Kevin had undergone as a child. As far as he knew it was last remaining. He ordered Reva to show him where it was, and had Jessica dig it up for him. Once he had it in his possession there was no need to keep Reva, and he ordered Jessica to get rid of her. Jessica killed Connors, and in that moment something changed. She broke free of him.
It was Kilgrave's first real taste of defiance.
He didn't have long to puzzle over it. Jessica ignored his calling and continued to wander away from him, causing Kilgrave to pursue her, and miss the bus skidding out of control.
He should have died. He abducted a doctor-David Kurata and forced Jack Denton to donate both his kidneys to him. Kurata was forced to perform the surgery on Kilgrave without any anesthesia and saved his life, while Denton had a stroke from the shock of losing his kidneys. Any records of his survival was buried and erased, and Kilgrave retreated to recover.
She'd broken free. She'd denied him. No one had ever done that. And while Kilgrave could have anything he'd ever wanted, there was something exhilarating about being refused. She needed to be punished,, obviously, but on his good days Kilgrave imagined winning her over proving to her that he loved her, and things returning to how they were meant to be, only this time there would be no denying that she was there because she wanted to be. Other days he dreamt about killing her, breaking her, making her love him, beg for him only to be denied by him.
Either way, Jessica needed to be dealt with.
Powers (exerts from wikia)
Mind Control: Kilgrave has the ability to control another person's mind through literal verbal commands, thanks to a virus that he emits through micro particles in the air. He controlled some for minutes, others for months. However, Kilgrave himself must be present for his commands to have an immediate effect; speaking over a phone does not enthrall his victim. Kilgrave's power also has a time limit of ten to twelve hours before his victim began to regain their will. Until the time limit is up, the victims will continue to follow Kilgrave's command unless he gives them a different order to follow.
His power is a side-effect of the viral treatment used to cure his childhood degenerative disease.
His thralls must fulfil Kilgrave's words specifically, but sometimes they are open to interpretation .