Why?
Before getting into the question of how you can teach your children to read and understand the King James Version, we should probably first discuss why you should want to do so.
If you are an English speaker, even if you are an atheist you should want your child to be familiar with the great works in the history of the English Language, and the King James Version is certainly close to the top of the list, if not at the very top.
Even some of the greatest skeptics were of this opinion:
“It is the most beautiful of all translations of the Bible; indeed it is probably the most beautiful piece of writing in all the literature of the world.” -H. L. Mencken
“The translation was extraordinarily well done because to the translators what they were translating was not merely a curious collection of ancient books written by different authors in different stages of culture, but the Word of God divinely revealed through His chosen and expressly inspired scribes. In this conviction they carried out their work with boundless reverence and care and achieved a beautifully artistic result.” –George Bernard Shaw
"It is written in the noblest and purest English, and abounds in exquisite beauties of mere literary form." -Aldous HuxleyThe influence of the King James Version on the English language has been huge, and there aren't many other texts that would be comparable in that regard.
It also happens to be a very fine translation. It is not perfect, but it has many advantages over most other options. See: An Orthodox Look at English Translations of the Bible.
Isn't it too hard?
For many centuries, even poorly educated people read and understood the King James Bible, because they made the effort to do so. For the most part, the King James Version is perfectly understandable for a modern reader. There are perhaps a hundred words or so that one would have to acquaint themselves with, if they were not already familiar with them. All of these words are found in a standard dictionary, and the intended meaning of the word in question will usually be listed as the primary or secondary meaning. There are also some handy guides online and in print that provide quick definitions with these words. And you could always look up a difficult text in the New King James Version, for clarification.
How?
First off, you have to teach them how to read, and teach them to love reading.
My wife and I home schooled our children, and the single best text we used was a book entitled "Teach Your Child to Read in 100 Easy Lessons," by Siegfried Engelmann. It provides a parent with simple and clear instructions as to how to use the text, and it effectively teaches a child how to read phonetically, and also how to understand the many quirks we have in English spelling (something that is often not taught in public schools in our times). Most importantly, it works. I started teaching both of my children how to read with this text when they were three, and had them reading on a basic level within a few months.
The first books I had my children read were in a series of Bible story booklets from Concordia Press that are designed for beginning readers -- the closest thing that they have to what we used in print now is in a series called "Hear Me Read.".
You should regularly read to your children. For very young children, I found reading them stories that rhymed got their attention, and so I read them rhyming Bible stories. Concordia Press has a large collection of short Bible stories that rhyme -- many of which I remember from my own childhood.
As they got a bit older, I read them a comic book collection of Bible Stories (The Picture Bible), and as their reading improved, they would read it on their own. This gave them an overall understanding of the Bible in broad strokes, and helped to improve their own reading.
In addition to reading books directly connected to the Bible, reading other classic texts to your children helps to develop a love for reading.
We did not have our children read much of Shakespeare, because Shakespeare's plays were not meant to be read -- they were meant to be watched. We had them watch all of his major plays -- some in multiple versions, and they enjoyed them. And this helped to familiarize them with Elizabethan English, and in a way that was not at all tedious.
Finally, when their reading level got to the point that they could begin to do it, I had them read the Bible to me. This helped their reading and pronunciation, and it also gave me a chance to explain any words that were obscure, and to discuss the meaning of the text. We started with Genesis, and stuck to the narrative portions of the Law and the Historical books. We eventually brought in the Wisdom books, the prophets, and also the Gospels and Epistles.
A very important help to this whole process was to get an edition of the King James that had modern spelling, punctuation, and paragraphing -- and to have the same edition in everyone's hand, so we were literally all on the same page. At the time, we used the Third Millennium Bible, but what I would recommend now is using the Cambridge New Paragraph Bible with the "Apocrypha". This edition is laid out in a way that is much easier for contemporary readers, and the more I use it myself, the more I have come to like it.
One other thing I did was to have my children memorize the names and order of the books of the Bible, and then we would do something which I learned from Sunday School as a child -- "Sword drills". When we finished reading the Bible, I would call out random Scripture references, and we would see which child could find it first. This taught them how to navigate their way around the Bible.
For more information:
An Orthodox Look at English Translations of the Bible
A Simple Approach to Reading the Entire Bible
King James English and Orthodox Worship