In Act II, Scene 3 of Proof, Claire is acting out of genuine love and concern for Catherine. Her motive is to protect Catherine and to try to find answers about Robert's genius, a genius she recognized but never understood, and discover if Catherine is delusional.
Claire fears that Catherine is emotionally and mentally unstable, suffering from the same propensity to mental disorder that their father had. Claire knows that Catherine is also a genius but doesn't understand it in Catherine any better than she did in Robert. Claire is herself brilliant but in a practical and down-to-earth way. This is why she spent fourteen years working extra-long hours to pay off the mortgage on the family home.
Claire is angry with Hal for what she sees as taking advantage of Catherine's unstable state. Catherine is in mourning; she is faced with difficult decisions; she is in mourning; she seems to be delusional about the notebook.