The chart looks as though you don't have to mentally switch colors, so you just do it as the chart shows it. I suppose you know that you are double knitting.
If the square on the chart is grey, you knit with MC, then purl the next (corresponding) stitch with CC, if the chart shows green square, you knit with the CC, then purl with the MC.
(Normally, you have to switch colors of chart from one row to the next, so even if it said "green square" on the chart, you would do "grey square" instead. But this chart doesn't seem that way, which is less confusing.)
Does this make things clearer?
The ways of Yarn are inscrutable.
I'm NOT a "miss", I'm a lady!