General Fusion has a neat approach to achieving nuclear fusion. By using large pistons, they will send a sonic shock wave from the edge of a sphere toward the center, where some fusion fuel lies. The way that these shock waves interact create an amplified version of the wave at the absolute center. Apparently, this amplification is so great that even conventional machinery can create pressures and temperatures so great that nuclear fusion can happen. We're looking for a much bigger boom here. So what if we start out with thermonuclear fusion? Oh my.
Could we create fusion at the center of Saturn? No. The center is made of rocky materials. Iron can fuse under certain conditions, but we would rather not go there. Additionally, the phase changes might present a barrier to the wave propagation, as waves tend to partially reflect at density boundaries. Also, the waves will somewhat diminish in intensity traveling over such long distances.
So where could we create fusion? As deep as we can go! Human ingenuity can create submarines that can go deep to pressure levels of 3000 psi or greater. The electronics and other systems probably won't need to be in a pressure vessel anyway. The higher pressure will affect the dynamics of the shaped charges, but they could affect it for the better (I mean, higher yield). An advanced society might be able to go much deeper. Actually, due to the lower molecular mass, the scale height is much larger than on Earth, maybe 60 km. You could easily go 400 km or more below the 1 bar level. Liquid Hydrogen should exist around 1000 km, and this will be even better for our fusion.
You need to get greater depth because in order to actually blow up Saturn, you need reaction mass to push off against. A wall of 100 km of relatively high density gas is okay to push off of, but not fantastic. You'll need a big boom for this blast to actually decimate the rest of the planet.
Using the General Fusion approach, we will arrange thermonuclear weapons in a large sphere, perhaps 100 km in diameter, deeper than 500 km depth. By the way, we will need a lot of bombs. They should also be high yield. Also, they need to be timed with absolute perfection so that the blast waves all culminate in one giant fusion ball in the center. Doing so will require correcting for the pressure gradient too, but I consider this challenge similar to what General Fusion is already dealing with.
We need to produce, at minimum, the dissociation energy of Saturn. This won't be enough to turn the planet into a cloud of gas (due to the specific physics of the explosion), but it might be enough to satisfy the OP. Fusion tends to release less than 1% of the $mc^2$ energy of matter, but the specific reactions are complicated. You might seed the fusion area with some Boron, Tritium, and other stuff to keep it interesting. But our goal is to have this area at such a high pressure and temperature that it becomes a "Mr. Fusion", combining whatever atoms that release energy, and even some that don't.
Optimistically, we would need a fusion volume of a box about 2,800 km on the side. This is about equal in size to the practical limitation of our giant thermonuclear fusion trigger spheres. But only a small fraction of that volume (at the center) will actually fuse.
So we'll need several of these spheres. Also need to make sure they all trigger at the same time. Best case scenario, you'll need a few 100. Each one of them will probably release more yield that all nuclear weapons that humanity has ever built. But the fusion that they trigger at the center of their spheres will buy you many orders of magnitude beyond that.
To recap: We will start with thermonuclear weapons. That means that a fission bomb will detonate, and its radiation pressure will detonate fusion fuel next to it. This happens simultaneously for many bombs arranged in a large (several 100s or 100s of km) sphere within the top of Saturn's liquid Hydrogen layer. These trigger a sonic wave which amplifies at the center of the sphere, creating a massive fusion ball which releases many times more energy than the original bombs. Now, this giant fusionsphere happens in 100s of places in Saturn's liquid Hydrogen layer in one hemisphere. The energy released is enough to blast the planet to bits, but most of the energy goes to releasing gas into space at high speeds. The outer layer above the bombs is entirely thrown out into space. This could provide impulse to push the remaining core out of the solar system or somewhere else. Generally, the solid core will not be blown away, but most of the light elements probably will be.
While grand, our solar system is almost certain to have the necessary fission and fusion resources to accomplish this. Substantial technology hurtles still remain, but president gives good confidence that they can be solved by an interplanetary society. The scale is a few orders of magnitude beyond human cold war activities, but this doesn't sound like a problem either.
Historical thermonuclear bombs have, in fact, used liquid Hydrogen. Saturn's liquid Hydrogen layer isn't the right isotope, but it might not matter, and you might be able to refine Deuterium out anyway.