Where are you publishing? In the US, the KJV is in public domain and there are no restrictions. But in the UK, the translation is owned by the crown and published by Cambridge, and there are restrictions and notification rules similar to those for other translations.
For most books, you do not have to seek special permission anyway unless the Bible is very heavily quoted; you are only required to give notice on the copyright page that you are using that translation. Typically, for example, you'll be allowed to have up to 25% of your book be Bible quotes as long as you don't exceed 500 verses (some copyright holders are even more generous, and a few are less generous). Check the version you plan to use; the limits and the wording for your copyright page are usually printed on their copyright page. For the KJV in the UK, the limits are no more than 25% of the book, not to exceed 500 verses.
Finally, I would suggest that even if you hope to publish in the US using the KJV, so that no restrictions apply, you still note on the copyright page that you are using the KJV. Some readers might like to know, and it is a customary courtesy to acknowledge any borrowed material.