I'm new to the topic too, so I'm just throwing in my 2 cents.
Quite simply put, if the core is made sufficiently dense (highly compressed mass is the key feature essential for causing gravitational contraction) with ionized hydrogen and some helium very quickly, you should get a plasma like medium at a decent temperature of a few thousand Kelvin, which sounds about right for nuclear fusion to begin. Adding more hydrogen to the surface would not trigger this. You'll need to inject high temperature hydrogen(deuterium to be precise) into the core at a very high pressure to make this idea conceivable. Since we're starting off with Jupiter here, the majority of the planet, except for the parts close to the core, is at a temperature too low to sustain nuclear fusion even though the nuclear fusion takes place in the
core, since any high temperature spark in the core would immediately be cooled down by the surrounding mass of gas, and as a result, the core would rapidly cool and expand, pushing on the gaseous surface and just increase its diameter. Depending on the scope of imagination that you would allow, if some isotopes of heavy elements with comparatively low binding energy per nucleon were compressed in the core(something of an atomic bomb), the series of initial nuclear fission reactions would spread outwards from the center of the core, eventually spreading all across the planet to the surface, converting it into a radiation ball and heating it up to allow chain reactions to continue. This would allow the core to sustain nuclear fusion and create enough outward radiative pressure to counterbalance the gravitational force pulling the mass in, and since Jupiter is already rather massive as a planet, the critical mass needed for sustaining the chain sequences should be met with the injected mass, adding to the bulk at the core. Compression and reduction in surface area to volume ratio of the radioactive parent nuclides increases the degree of chain reactivity, a concept used in nukes.
Summarizing it, you'll need to inject super-compressed high temperature deuterium into Jupiter's core together with some Helium and heavy isotopes with low binding energies per nucleon. Remember that these actions are adding mass to the planet too, and the reactions should theoretically convert Jupiter into a star, although it's practically impossible.(As of yet

)