Julien does a good job as a tutor, but he can't get over the inequality between the mayor's house and the kind of poverty he (Julien) grew up with. He thinks it's unjust and harbors a secret hate for the mayor.
Some days later, Julien runs into his brothers. They beat him up out of jealousy and leave him lying half-conscious on a path. Minutes later, Madame de Rênal finds him while walking and thinks he's dead.
Julien finds Madame so beautiful that he actually dislikes her. He assumes that a woman that beautiful could never be a good person.
Meanwhile, Madame de Rênal's chambermaid Elisa falls totally in love with Julien. Julien overhears her one day arguing with a male servant, who apparently used to have his eye on Elisa before she decided she liked Julien more.
This just makes Julien pay even more attention to his appearance, since he's a proud kid.
Madame de Rênal wants to improve Julien's appearance even more by giving him clothes as gifts. Her husband thinks this is a silly idea, though.
As time passes, Madame becomes more and more affectionate toward Julien.
One day, Julien walks in on Madame while she bursts into tears. He asks what's wrong, but she just says she'd like him to go for a walk with her and her children. She takes his arm and leans against him.
She tells him that she's like to give him some money for new clothes. She adds that she would prefer for him not to tell her husband about it.
Yup, you can probably see where this relationship is headed.
Julien surprises us by rejecting Madame's offer and saying that he won't dishonor his boss by taking gifts behind his back.
Madame goes to her husband and tells her about what happened. He can't believe that a servant would reject an offer like hers.
He meets with Julien and forces him to accept a gift of a hundred Francs.
Madame takes Julien for another walk one day and leads him to the town bookstore. She buys a bunch of books that are supposed to be for her children, but she knows they're all ones that Julien wants to read.
Julien convinces the mayor to take out a subscription to the bookstore through one of his servants. This will give Julien access to all the books he wants.
All this time, Madame de Rênal thinks constantly about Julien.