When you connect an external display to the chromebook it will switch its resolution to the 'best available' that the external display can handle. If you plug in a screen with a lower resolution than the chromebooks own display, both will drop down to this level - that is why you're getting the black bars on the CB screen. If you plug in a screen with a higher resolution than the chromebooks own display, the CB screen will go blank but you should get a great picture on your external display. The big difference between Chromebooks and other OS is that there are no controls to change the resolution - it does the best it can and you have to like it!
I suspect the issue is with your TV, in that it is asking for the wrong resolution and then stretching the picture to fit.
I'd suggest trying your CB with other TVs, and trying other computers with your TV to isolate the problem. It is also worth exploring your TV's menus to see if you can change the resolution it asks for, and check your TV's documentation about what resolutions it supports.