Let me first say that some kind of magic would make this much easier. The conditions required to initiate nuclear fission or fusion are not likely the sort that would develop readily in Earth-like creatures. I believe that your best bet is, as you seem to be considering, making the dragon a space-dwelling creature. Perhaps it originated as a terrestrial organism that used solar energy to propel itself, storing the energy for rainy days. If it were quite large (but ideally thin), and virtually covered in solar panels, and had sufficiently good batteries to survive dark periods, that sort of thing could happen. Of course, a creature whose skin can absorb solar energy more efficiently than a solar panel, and whose body can store it more effectively than current batteries, will probably not resemble Earth life, in terms of its genetic code, the molecules that make up its body, or its overall appearance. It might have a lot of silicon in its skin, for example.
Now, a creature that can survive on solar energy might not have to breathe, which would be ideal for navigating space. I simply do not think that nuclear fission would evolve first as an energy source, even in the presence of ideal fuel, because the energy barrier for starting it is pretty high, relative to biological energy scales. But a creature that could extract huge amounts of energy from solar power might be able to overcome it. Once the "dragon" can produce solar energy, it might be able to evolve the ability to use nuclear fission (assuming that nuclear isotopes are common on its home planet) as a backup energy source. How a creature would evolve a controlled fission mechanism without blowing itself up, I cannot imagine. Even if such a mechanism managed to evolve, the dragon would need a heavily shielded inner fission chamber, to protect its batteries and solar cells from the radiation and heat. Maybe it could release the heat by breathing "fire"? Again, to have any chance of dissipating the heat produced, the dragon has to be pretty large. But this makes flight even more unlikely, and means that the dragon probably has to have bones made of something pretty exotic. Perhaps your planet has very weak gravity (very small). But then it will not have an atmosphere, which means the dragon would have difficulty flying. Perhaps your planet also has some sort of fuel, and the dragon flies by expelling it at high speeds, perhaps using a mechanism powered by its solar intake (or by the fuel itself). This would also help it get into space.
Then we have the problem of how its solar panel skin can survive the descent back to its planet from space. Maybe it has a layer that it can extend for protection, although again, this is not going to be anything like what we call a biological material on Earth. Once in space, it can collect even more solar energy, but has to be careful not to run out of whatever propellant it collected on the planet. It might also be able to use solar sails to change direction, although it would likely only evolve these after living in a space environment for a while.
But what is more, you want it to travel between solar systems! I think the best bet for this creature, if its brain is also silicon, is to shut itself down, put its brain in deep-freeze, and reactivate only when it nears another star, or when necessary to adjust course. Presumably this creature is highly intelligent, if it can figure out, in its head, the proper trajectory to a distant star system.
Of course, what you end up with is a dragon built like something from Transformers: Age of Extinction, but you probably realize that the best way to have a nuclear reactor spaceship dragon is to have a nuclear reactor dragon spaceship.
