The problem is that once he's there Ripley falls in love with Greenleaf's life himself. Ripley is superficially charming but isn't the sort of person you like if you know him in a profound way, so he decides to kill Greenleaf and assume his life because they look so similar physically.
From there Ripley has to use his skills as a chancer to convince:
1) People he's Greenleaf, so he can have Greenleaf's money.
2) The police that Greenleaf is safe (and then up to no good when he can't live as Greenleaf any more).
3) Greenleaf's family and former love interest that he really cares about Greenleaf.
So why do I feel this book is so dated? Ignoring the fact that there is no forensic trail for the Italians to follow, or the fact that there are no computers, what is it first thing that the police do these days when someone goes missing? They investigate the last person to see him. That doesn't happen here. Ripley is taken at his word all the time, when even the simplest investigation of him would have shown him to be a liar. That irritates me no end. To my mind the book might be better described as The Fortunate Mr Ripley because he certainly road his luck in this book.
If you can live with this issue read this book, you'll like it. If you can't, I probably wouldn't bother with this book. It might irritate you.