This is an excellent site to look at the first approved Catholic translations, and it’s worth noting that the Rheims New Testament came out before the King James Version (the Douay Old Testament took longer). There are a number of differences between this and the more common Challoner revision, from which most modern Douay Rheims Bibles are produced. Sometimes this is the more literal to the Latin and Greek, and perhaps the most literal ever made.
Here’s an example from Matthew Chapter 21:
28. But what is your opinion? A certaine man had two sonnes, and comming to the first, he said: Sonne, goe worke to day in my vineyard. 29. And he answering, said: I wil not. But afterward moued with repentance he went. 30. And comming to the other, he said lifewise. And he answering, said: I goe Lord, and he went not: 31. Which of the two did the fathers wil? They say to him: The first. Iesus saith to them: Amen I say to you, that the Publicans and whoores goe before you into the Kingdom of God. 32. For Iohn came to you in the way of iustice, and you did not beleeue him. But the Publicans and whoores did beleeue him: but you seeing it, neither haue ye had repentance afterward, to beleeue him.