Catelyn Stark hated her husband's bastard son. Not because Eddard had cheated on her and bedded another woman but because of her much his bastard looked like Ned. His true born sons did not even look like him as much as Snow did.
Snow had this highborn look about him without even trying, the high cheekbones, skin akin to his last name and black hair that was as dark as midnight. It made her wonder how the boy's mother was, as even though Ned skin was pale it was no match to how pale and snow like Jon's was.
Catelyn had once, long ago, asked her husband who the boy's mother was but he had asked her to never speak of it again. She had pointed out that she was his wife and had a right to know what woman he had bedded, the woman he broke his marriage vows for. Ned had been furious at this and told her that she had no right to ask about Jon's mother again that she had no need to ask about what happened at war.
This is why Catelyn hated Jon Snow and what he represented. She did not know why Eddard and their children, except Sansa who at one point had hated Jon as she believed that he was beneath her as she was a proper Lady but she had only believed this as that was what her Mother had told her and Arya once at one of their lessons together but once she had voiced this to her oldest brother he had said that didn't matter and he was still her brother no matter how his mother was, treated Jon like he was a trueborn stark.