While you can't turn Jupiter into a star, it is not ruled out that you could turn Jupiter into a catastrophic thermonuclear bomb. The limitations to this was calculated at Lawrence Livermore in the 1970s, as a continuation of the work done to check to make sure Earth's oceans wouldn't ignite due to the deuterium content of water. Necessary Conditions for the Initiation and Propagation of Nuclear Detonation Waves in Plane Atmospheres by Weaver and Wood, couldn't rule out a self-sustaining ignition shock-wave in a planetary atmosphere at a deuterium concentration of more than 1 percent at ordinary liquid densities.
Although this makes the oceans safe, Jupiter is big, and it might have segregated a deuterium layer deep inside which has a high enough concentration to allow a self-sustaining nuclear ignition. Then if you drop a configuration of plutonium designed to detonate the deuterium by a nuclear explosion at the appropriate depth, you could get a detonation wave that ignites the entire deuterium layer within a very short time, the time it takes a shock wave to encircle Jupiter.
The energy output could convert a non-negligible fraction of the deuterium in Jupiter to 3He/tritium, and release enormous amount of energy. If 1 Earth mass of deuterium is ignited by the ignition shock wave, the energy release is 1038 J, over a very short time, perhaps an hour or two and this is already 10,000 times the energy output of the Sun in a full year. The resulting explosion would destroy that part of the world facing Jupiter, and probably bake the rest. I don't lose sleep over this, though.
If there is a natural trigger for such an explosion, perhaps the collision of a rocky planet with a gas giant, one might experimentally observe such planetary mini-supernovas somewhere. This was suggested in section VIII of Weaver and Wood's paper.
How opaque is that -- would we be able to see a couple of meters, some kilometers, or nothing at all?
The photosphere of our sun is somewhere on the order of 500 km thick. For a quick ballpark, you can imagine an exponential decrease in the transmission of light which about this characteristic thickness. It might be a little less, but it's still order-of-magnitude accurate. That means that you would (theoretically) be able to see local objects just fine within the photosphere.
It's difficult to compare this to Earth's atmosphere, because scattering of light is both Raleigh and aerosol, and the concentration of aerosols differs at different times. However, even Raleigh scattering attenuates more than half of the light within 100 km. That means that you can actually see better in the sun's photosphere than you can in Earth's atmosphere.
Would the matter directly in front of our cockpit window be bright, or dark?
Now for the qualifier, being inside the photosphere would render human eyes useless instantly, because it would be like painting the sun's brightness over your entire field of vision. A spacecraft 100 km away could be identified with a telescope (if this telescope is not actively melting), but it would not be any brighter than the background of photosphere gas. Nonetheless, it would still be identifiable by color, contrast, and so on. In fact, if the spaceship was mostly absorbing of radiation, it would be a distinct black spot among a bright background.
I could get more specific to speculation about spacecraft. They will need to have stored thermal mass in order to absorb more heat than they emit. Reflection is a good option, but it's not perfect. Active cooling of the spacecraft's surface is likely in order to avoid melting. Thus, spacecraft would be slightly "camouflaged" by thermal necessity, but not perfectly.
Does this opaqueness only apply to a certain layer (i.e. would we be able to "see" again once we go below the photosphere), or are all lower layers of the sun opaque?
Your vision will certainly decrease as you reach lower layers - since they have higher density and give off more radiation. The photosphere is only unique in that the radiation emitted commonly makes it out into space where it can continue indefinitely. For the lower convection layer zones, radiation is given off at a much higher rate, but it is almost entirely reabsorbed.
Stars above a certain mass threshold actually transmit their heat through the interior zones through a radiation gradient. So I hope this helps to illustrate that the inner zones are not "less bright". They would be blindingly bright (well the photosphere is already blindingly bright, but these would be worse), but possibly at higher temperatures, which go beyond the visible range.
Best Answer
Jupiter's mass is too small to produce nuclear fusion.
This wikipedia page explains the detailed requirements of nuclear fusion:
http://en.wikipedia.org/wiki/Nuclear_fusion