The King James Version has 39 books in the Old Testament, all written in Hebrew. However, the Old Testament of the New American Bible contains 46 books, including an additional seven (Tobit, Judith, 1 & 2 Maccabees, Wisdom, Ecclesiasticus, and Baruch) which were part of an ancient Greek translation of the Hebrew Bible called the Septuagint. Those are the main differences. So if you want to read those books you’d need to read those in the NAB which is the version Catholics use at mass.
Other differences is that the NAB was translated in 1970 — more than 350 years after the KJV was produced — which makes it more accurate academically. However, the New King James Version was produced in 1982 and it has included these newer findings of the biblical research field.
So, any translation is ultimately an interpretation. The translator will coma this from a variety of angles. Some choose to translate faithfully word-for-word to the original document, while others will communicate the essence of the passage faithfully. Neither is necessarily more “correct.”
So yes, a Catholic can read the KJV, but the approved version for Catholics is the NAB.