Double cloth is two layers of fabric stitched/tacked together throughout, so that you have a double-thick fabric. I've only ever seen it in wool - I don't know if this fabrication is possible in other types of fabric. It's sewn together with what, essentially, is a fancy version of a flat-felled seam.
Think of this as attaching 4 layers, but you sew 3 of those layers together, then hide the seam allowance between layers 3 and 4. Clear as mud?
Once you've cut your pieces, you have to decide which way you will be pressing your seams - i.e. to the back or to the front. I chose to the back. I had to peel the fabric of the back piece from the edge to 1/4 inch beyond the seam line. To make this easier, I basted at 7/8 inch . Only later in the process I realized that basting with a bright coloured thread actually made it easier to see when I had finally reached the basting line, because you can see that stitching between the layers of fabric.
What really had me puzzled was how to deal with the ends of seams that would then be turned in on the edges of the coat. I figured it out when I got there. You have to peel both edges right at the very end, to give you the two layers on the edge that can be turned in for the "hem".
The hems have to be stabilized somehow, or they'll surely stretch. I looked for very narrow twill tape, but couldn't find anything narrow enough. What I found in my trims box, was silk knitting ribbon (now where did that come from???) It's very narrow - just what I needed, and very thin - no unnecessary bulk, and very stable - also what I needed.
If you want professional pictures of the process, in pretty colours - not my blah grey, check back to Issue 123 of Threads from February/March 2006 - there's an article by Anna Mazur on this wonderful fabric. (Page 60 - to be exact.)