Both the Douai Rheims (or Douay Rheims)) and the King James Version (KJV) are English versions of the Bible, and the Biblia Sacra iuxta Vulgatam Versionem is a Latin version of the Bible. They all originally were translations made from older languages into the common language of their time and location.

The Douai Rheims was a 1610 translation into English done originally from a glossed/annotated 16th century Latin text of the Vulgate, with some consultation of available Hebrew, Greek, and other language manuscripts. It was revised in 1635 and again in 1750. It is an English translation from a Latin translation (the Vulgate) that was itself a translation from Hebrew and Greek manuscripts. It would be analogous today to an Arabic book that was translated into Chinese, and then that Chinese translation was translated into English. As is commonly known, much is lost or changed in translation.

The KJV was a 1611 translation into English from generally sound Hebrew Old Testament texts but considerably flawed Greek New Testament texts. A few parts of the KJV's flawed Greek texts were translated into Greek from a glossed/annotated Latin text of the Vulgate, which contributed to the Greek text's (and KJV's) corruptions. The KJV was revised many times in its first 160 years, with the most major and permanent revision of spelling and punctuation done by Benjamin Blayney in 1769.

The Biblia Sacra iuxta Vulgatam Versionem is the current version of the Latin Vulgate that was originally begun by Jerome in 382 A.D. and completed in 405 A.D., with the intent to provide a standard Latin version of the Bible and eliminate the differences in older Latin texts. It was many years later that Jerome's work was bound in a single volume and called the Vulgate. Jerome did not do all of the translation in that volume. Only the Old Testament, Tobit, Judith, and the four gospels can definitely be attributed to Jerome. Over the centuries Jerome's text continued to be copied, annotated, corrupted, and revised. The Biblia Sacra iuxta Vulgatam Versionem is the best, publicly available, current reconstruction of Jerome's original work.

So which of the three translations is most accurate? It is almost a moot point. Today there are much better Roman Catholic and Protestant English versions. The New American Bible (Roman Catholic, book review) and the English Standard Version (Protestant , book review) would be my choices. They both are far more accurate and much more readable for modern readers than the Douai Rheims or the KJV. As for the venerable Vulgate, Latin is a dead language, and the original purpose of the Vulgate in the 4th century was to provide a Bible translation in the Roman Empire's common vernacular of Latin. Beyond liturgy and liturgical studies, the Vulgate's primary value today is to textual scholars who use the Vulgate along with many other ancient manuscripts in Hebrew, Greek, Syriac, Coptic, and other languages, to help determine the wording of the original biblical texts in Hebrew and Greek. Frankly, instead depending on the Vulgate, today's Roman Catholic and Protestant scholars doing biblical research, interpretation, or modern translations depend on critical editions of the United Bible Societies' (UBS) Hebrew Old Testament text (Biblia Hebraica Stuttgartensia, 5th edition) and the Greek New Testament (Nestle-Aland, 27th edition). The Hebrew and Greek text available together in one binding in the UBS's Biblia Sacra Utriusque Testamenti Editio Hebraica et Graeca.

More info on the KJV is available at http://www.bibletexts.com/kjv-tr.htm.

More info on the Vulgate is available at http://www.bibletexts.com/glossary/vulgate.htm.