Just to make sure, do you mean you want the header (with logo block etc) having a different background color when viewing a private conversation?
I would think the easiest way to do that is to add
background-color: red !important;
This is with the disclaimer that I understand the XF template system correctly and that css will only be called when viewing a conversation. If you want to do it with extra.css you will most likely have to use conditionals.