Double knitting pretty much requires you to read and knit from knitting charts, but one potential problem for knitters new to double knitting is the concept that double knitting charts look just like regular knitting charts, only each box represents two stitches instead of one.
To help you understand how to read a chart for double knitting, lets take as an example the Peace Sign Chart shown in the picture. This is a great choice for someone new to double knitting because it's symmetrical.
If you look at the first row of the chart, you'll see that if you were working regular two-color knitting you'd knit 12 stitches in your first color, 7 in your second color and 12 again in your first color.
But in double knitting you are alternating stitches on each side of the work as you knit across. So those first 12 stitches are actually 24 loops, 12 for the "front" or the side facing you and 12 for the "back" or distant side. Because you want those stitches to be your background color, just knit the "front" stitches with the "front" color and purl the "back" stitches with the "back" color 12 times, for a total of 24 stitches.
Now you need to switch colors to start forming the image. The way that is done is to simply knit with the "back" color on the "front" stitch and purl with the "front" color on the "back" stitch as many times as you need to for the pattern. So here you'd knit 1 with the back color, purl 1 with the front color 7 times, for a total of 14 stitches.
Then you just switch back to knitting the front with the front background color and purling the back with the back background color for the remaining 12 stitches on each side.
To work the second row, you do the same thing, but as with regular two-color knitting, you have to read the chart from left to right on "wrong side" rows instead of from right to left. It just so happens, though, that this chart has solid-color rows on the "wrong" side (because it was made for beading) so that makes things even easier.
Of course not every chart will make it that easy for you, and you do need to select your motifs wisely when it comes to color knitting. You're producing a mirror image on the "back" side of your knitting, so that means if you are knitting letters they'll come out backwards on the back side. It's better to stick to motifs that are either symmetrical to start with or will look good when reversed, too. An example of that second one might be the umbrella chart. The umbrella hook will face the opposite way on the back as opposed to the front, but that's no big deal.