I'm going to have to say no, you can't improve the video quality of your cable tv by running it through the computer, if that's what your asking. Not to mention you would also need a tv tuner card, as the 5870 cannot do input.
A video card will not improve video quality from a TV tuner. You need to buy a higher quality TV tuner or switch to a provider that has better signal quality. Of course if the source or master video quality is poor then there's not much you can do there.