In The Talented Mr. Ripley, Patricia Highsmith creates a compelling psychological drama precisely because she does not explain how Tom Ripley becomes a killer. Instead, her third-person narrator provides just enough information at every step to enable the reader to see how each action would make sense from Tom's perspective.
Tom is a sociopath who believes that his own needs and desires are paramount. Coming from a background of poverty and deprivation, he lives his own twisted version of the American Dream. Tom wants the finer things in life, but he has a distorted sense of what constitutes the hard work by which he will achieve those things. It is likely that he never had any regard for human life or real fear of the consequences of his behavior. He suffers no moral qualms when he breaks the law.
While Tom knew that Dickie was rich, it is not until he experiences the way he is living in Europe that he understands what "rich" really means. A whole new world of desires is opened up, and Tom feels fully entitled to position himself to take full advantage of those possibilities. And surely, he rationalizes, he would benefit from them more than Dickie does.