1. Check to see if the KJV has a marginal note that agrees with the modern translation or vice versa. If so, it means the original text is subject to more than one interpretation. The KJV translators chose one option whereas modern translations have chosen another. For example, John 14:18 has ‘comfortless’ in the text and ‘orphans’ in the margin of the KJV, whereas modern translations have ‘orphans’ in the text.
2. Check to see if the English words used by the KJV have changed their meanings. Use the Oxford English Dictionary online to do this. It may be that the KJV means the same thing, but our use of English has changed enough that we wouldn’t realize it. A great example is the word ‘conversation’ in Phil. 3:20. In 1611 it could mean ‘the action of living or having one's being in a place or among persons,’ which is much closer to the meaning of the Greek word politeuma = “citizenship, commonwealth” than the meaning of the modern English word conversation.
3. Check to see if the textual basis of the KJV is different from that of the modern translation. Checking a technical commentary is one of the best ways to find this out. Examples include the Baker Exegetical Commentary, Zondervan Exegetical Commentary, NICOT or NICNT. In the OT, the KJV sometimes follows the Septuagint (Greek OT) rather than the Masoretic text (Hebrew OT). For example, Prov. 18:24 – ‘must shew himself friendly’ (KJV) and ‘may come to ruin’ (NASB). In the NT, the KJV was translated primarily from the 1550 edition of Stephanus Greek NT. Most modern translations use a critical Greek text which tries to reconstruct the original text of the NT based on the over 5000 manuscripts available at the time of their translation. An example of this is in Eph. 5:9 - ‘fruit of the Spirit’ (KJV) and ‘fruit of the light’ (NASB).
4. If 1-3 aren’t the case, then it’s possible that we have learned more about the biblical languages and so better understand what the authors were communicating than we did in 1611. For example, in Prov. 27:16 the KJV uses the word ‘bewrayeth’ which meant ‘reveal, expose, discover (unintentionally, and usually what it is intended to conceal).’ Modern translations have ‘grasps.’ It appears that our understanding of the Hebrew meaning of yiqra’ in context has improved.