First of all, Catelyn doesn't care about that. It is an arranged marriage, and she only grew to love Ned over their years together. It is a common thing for noblemen to father bastards in their society, however out of character for Ned, so there is no reason for her to take it personally, as Jon was there practically for the whole duration of their marriage. He is not some late-come interloper, he was more or less part of the Stark family when she came to Winterfell, like Benjen. Catelyn's hatred of Jon is founded solely in his threat to her children's inheritance (and possibly their father's esteem and affection). She would not have cared, as she admits, if Jon were left somewhere with his mother. She hates that he is in Winterfell, without his mother and thus no competition for Catelyn, but a rival child of Ned's. The fact that Ned deals with Jon as he does plainly indicates he feels affection and paternal responsibility to the bastard, which could lead to givnig him responsibilities or privileges which might usually go to Cat's children instead.

The portrayal of Catelyn in the books is a consumate and obsessive mother. She lives entirely for her children, and though she loves Ned, that is a distant second to her loyalties to her blood kin. For the sake of her sister and children she throws Ned on the fire and is willing to betray his bannermen and his legacy to appease his killers in order to regain her children, when every single other person in the universe says that would be a stupid move. Her hatred of Jon as the mother of those he could challenge or threaten in the future is far more in character than as the wife of one whose past betrayal he personifies.