Personally, I would just play the DVD using the computer - you won't get the quality loss and resolution munging that happens when the signal is converted to analog at the DVD player then back to digital at the tuner.
Of course if you want to stick with the DVD player and tuner option (so you can use a remote control?), then you should try changing display settings in your DVD player setup menus, and also options in your TV tuner software, which probably has similar settings.
DVD movies are normally in a 16:9 aspect ratio (1.78:1, typical widescreen) or even 2.35:1 (Cinemascope). If your monitor is also widescreen, 16:9 movies should have little or no black border; with 2.35:1 Cinemascope movies (which is most movies now days), it is normal to have black borders of maybe 20% of screen height above and below the video. This gives the video the same look as in a cinema.
If we are misunderstanding you, please give a diagram of your monitor, showing how much video there is with the DVD player/TV tuner vs movies played back directly on the PC.