When it comes to binding off double knitting, you have a couple of different options. You can choose one color and bind off in the standard way, or you can do a simple two-color bind off.
All you have to do to bind off in two colors is knit the first stitch in the established color, move the yarns forward, twist as usual before purling the second stitch in the established color, then pass the first stitch over the second.
Move both yarns back, knit the next stitch, pass the first stitch over and so on until all the stitches have been bound off. Trim your yarns, put the thread that's the same color as the last stitch through that stitch and pull it tight.