Tom Ripley is a young man struggling to make a living in New York City by whatever means necessary, including a series of small-time confidence scams. One day, he is approached by shipping magnate Herbert Greenleaf to travel to "Mongibello", in Italy, to persuade Greenleaf's errant son, Dickie, to return to the United States and join the family business. Ripley agrees, exaggerating his friendship with Dickie, a half-remembered acquaintance, in order to gain the elder Greenleaf's trust.
Shortly after his arrival in Italy, Ripley meets Dickie and Dickie's friend Marge Sherwood; although Ripley ingratiates himself with Dickie, Marge does not seem to like him very much. As Ripley and Dickie spend more time together, Marge feels left out and begins insinuating to Dickie that Ripley is gay. Dickie then unexpectedly finds Ripley in his bedroom dressed up in his clothes and imitating his mannerisms. Dickie is upset, and from this moment on Ripley senses that his wealthy friend has begun to tire of him, resenting his constant presence and growing personal dependence. Ripley has indeed become obsessed with Dickie, which is further reinforced by his desire to imitate and maintain the wealthy lifestyle Dickie has afforded him.
As a gesture to Ripley, Dickie agrees to travel with him on a short holiday to Sanremo. Sensing that Dickie is about to cut him loose, Ripley finally decides to murder him and assume his identity. When the two set sail in a small rented boat, Ripley beats him to death with an oar, dumps his anchor-weighted body into the water, and scuttles the boat.
Ripley assumes Dickie's identity, living off the latter's trust fund and carefully providing communications to Marge to assure her that Dickie has dumped her. Freddie Miles, an old friend of Dickie's from Dickie's same social set, encounters Ripley at what he supposes to be Dickie's apartment in Rome; he soon suspects something is wrong. When Miles finally confronts him, Ripley kills him with an ashtray. He later disposes of the body on the outskirts of Rome, attempting to make police believe that robbers have murdered Miles.
Ripley enters a cat-and-mouse game with the Italian police but manages to keep himself safe by restoring his own identity and moving to Venice. In succession, Marge, Dickie's father, and an American private detective confront Ripley, who suggests to them that Dickie was depressed and may have committed suicide. Marge stays for a while at Ripley's rented house in Venice. When she discovers Dickie's rings in Ripley's possession, she seems to be on the verge of realising the truth. Panicked, Ripley contemplates murdering Marge, but she is saved when she says that if Dickie gave his rings to Ripley, then he probably meant to kill himself.
The story concludes with Ripley traveling to Greece and resigning himself to eventually getting caught. On arrival in Greece, however, he discovers that the Greenleaf family has accepted that Dickie is dead, and Ripley shall inherit Dickie's fortune according to a will forged by Ripley on Dickie's Hermes typewriter. While the book ends with Ripley happily rich, it also suggests that he may forever be dogged by paranoia. In one of the final paragraphs, he nervously envisions a group of police officers waiting to arrest him, and Highsmith leaves her protagonist wondering, "...was he going to see policemen waiting for him on every pier that he ever approached?"

Question: What does Herbert want Ripley to do?
Answer: Convince Dickie to return to his father.
Alternative answer: travel to italy to find and convince dickie to return to US