The menu bar on an inactive display (displaying a background window) is dimmed - grayed, if that's what you mean by "light black".
To get rid of it, my experience so far has been that the app has to be in full screen mode (as in an OS X full screen mode). Then, videos also played full screen within the app should fill the display (assuming a 16:10 display, you'll have black letter-box bars for 16:9 and wider aspect ratio video). Most of the time I'm in extended desktop mode, it's through AirPlay, but I believe the Display Preferences for the external display should be the same as wired.
If the display is a primarily a TV display, check the overscan settings in OS X. If on, you might be only getting part of the menu bar displayed. Some HD TVs also have their own settings for "overscan" to hide certain broadcast artifacts. This should be off, unless you actually see artifacts occurring regularly. Otherwise, the TV won't display the outer couple of pixel rows.