[ KJV: Psalm 23 / source ]
Like all Protestant Bibles (and Catholic ones), the fabled King James Bible has somewhat of a bias, based on the beliefs of the translators, but not overly so, in my opinion.
I myself use the RSV, which is a revision of the KJV (taking out the archaic language: "thee's" and "thou's," etc.). I used to use the New American Standard Bible (NASB) as a Protestant (which revised the American Standard Version [ASV], which revised KJV, into more "American" English), and first read most of the Bible in that version. The Catholic version of the RSV actually modifies very few passages: I think it is only 4-6, if that many. One of them is Luke 1:28: "Hail Mary, full of grace." That shows how little Protestant bias is perceived to be in the RSV, and by indirect implication, the RSV.
The thing about the KJV is the beautiful language and expression (who could not love, e.g., Psalm 23?). But, by and large, Protestant Bibles are not a problem, as much as false Protestant doctrines. I would concentrate more on those, and not the Bible translations, which can be criticized, for sure, but are not at the heart of Protestant-Catholic differences, by any means.
If you read the Douay-Rheims, be aware that it is a translation of a translation, too: of the Latin Vulgate, whereas most translations today, including newer Catholic ones, translate from the oldest available Greek and Hebrew manuscripts.
I suspect that most of the bishops of the Church (and the Holy Father) would rejoice that Catholics are reading the Scripture, whatever the version is. The main thing is to read the Bible. But there are "approved" versions, such as the NAB (personally I care very little for its style), and the RSV-Catholic edition. I have found many articles that discuss these issues or related ones:
Bible Translations Guide (Catholic Answers)
Choosing and Using a Bible: What Catholics Should Know, Fr. Ronald D. Witherup, Catholic Update, July 2004.
Choosing a Bible Translation, Jimmy Akin, This Rock, April 1994.
Bible Translations, Jimmy Akin.
Bible Versions and Commentaries (EWTN: Colin B. Donovan, STL)
Bible Translations (EWTN: Matthew Bunson)
Finding a Translation: How to Select a Catholic Bible, John Osman, Catholic Spirit, January 2007.
Choosing a Bible Translation, Katerina Ivanovna (Evangelical Catholicism site)
Are All Bible Translations Created Equal?: The Protestant Bias of the NIV, Steve Ray [this is a .doc file]
Which Bible Translations Are Best For Catholics? (Catholics United for the Faith)
Choosing a Catholic Bible (Adoremus Bulletin)
Uncomfortable Facts about the Douay-Rheims, Jimmy Akin, This Rock, February 2002.
Pius XII on the Authenticity of the Vulgate, Jimmy Akin, April 1994.
English Translations of the Bible, Felix Just, S.J., Ph.D.
Catholic Encyclopedia: Versions of the Bible [note: much of the article is about non-English versions, but there is a lot about the English ones, too]
Catholic Encyclopedia: Douay Bible