What does Leck almost do that reminds Katsa to kill him?
In Graceling, King Leck has a grace--which is basically a superpower--that allows him to use his voice to convince others to do whatever he wants. The moment Katsa hears Leck speak, she forgets that he is evil and cruel, and that he is out to hurt his daughter and many other people. Katsa knows something is not right, but she is too confused to remember that Leck needs to be killed.
In the end, Katsa's love for her boyfriend Po brings her back to reality. Po has a grace that allows him to see some of what happens inside other people's minds, but he has spent his whole life hiding the truth about this. If people knew about Po's ability, they would shun him or use him to gain power; he would become both an outcast and a tool. Katsa, whose ability to kill made her an outcast and a tool for many years, knows how painful such a life can be.
King Leck knows Po's secret, and he nearly reveals it to Po's whole family. When Katsa realizes what Leck is about to do, her mind snaps back to clarity. She throws a knife and kills Leck before he can speak.
Leck has the ability to manipulate people and their thoughts and when Leck worms his way to Po's family and tries to tell pos brothers and father the truth about pos misinterperatied gift katsa reaches for her dagger and without thinking launches it at lecks mouth and kills him instantly so that her lovers secret is safe from those who might want to use and abuse his(po's) gift