It depends on the wavelength you are using. The Milky Way galaxy is transparent to radio waves and to very high energy X-rays and gamma rays so at these wavelengths the Milky Way makes no real difference in our ability to see the distant universe over the full sky. At ultraviolet, optical and short- wavelength infrared, the amount of obscurration varies enormously. The Milk Way we see with our eyes obscures a band about +/-20 degrees or so to either side of the plane of the galaxy. This is a significant fraction of the full sky, and I think the calculation ( an integral has to be evaluated) yields about 30 percent of the full sky ( about 42,000 square degrees = full sky) is in some way obscured. Towards the edges of this 'Zone of Avoidance' you can see out to various distances depending on exactly your direction of view because the shape of the Milky Way on the sky is ragged. At infrared wavelengths, the Zone of Avoidance contracts a bit, but then at far-infrared wavelengths interstellar 'cirrus' dust clouds are just about everywhere on the sky at 50 - 300 microns wavelength and the amount of clear sky left shrinks to 20 percent or so. These numbers are all rather approximate and off the top of my head, but are qualitatively pretty reliable. If you want the clearest view nothing beats radio and gamma ray wavelengths.