I saw the film 10 Things I Hate About You. The plot is about Patrick Verona getting paid to love Kat so Cameron James can date Bianca Stratford.
But I don't understand why this title was given to this film.
The title is taken from a poem the main character Kat writes about her love interest in the film Patrick. In the poem Kat lists things she 'hates' about Patrick (also some things she loves) because she feels betrayed after finding out the reason behind his courting attempts. She also hates the fact that she still loves him despite his betrayal. There are also other meanings intertwined. An entry on the Wikipedia page for Karen McCullah (the films screenwriter) it states:
As a high school student, she maintained in her diary 10 Things I Hate About Anthony, her boyfriend at the time, which ultimately led to the title of her 1999 film, 10 Things I Hate About You.
@TG01 mentions in his answer below: "[The plot]...was based on Shakespeare's The Taming of the Shrew. So, '10 Things I Hate About You' is a pretty clever, rhyming homage to the source." which is a pretty interesting tidbit.
Here is the scene: