Volcanoes occur when molten rock from underneath the Earth's crust rises up. This molten rock comes from the mantle, not from the Earth's core. Because of plate tectonics, volcanoes can form relatively quickly.
The interior of the Earth is so hot that rocks are melted, and much of this heat radiates to the mantle as well. The mantle itself is so hot that efforts to drill into it have failed as drilling equipment melts when it comes into contact with it. The mantle is closer than many imagine; the Earth's crust is only about 35 kilometers deep, but its depth varies considerably.
When pressure in the mantle builds, molten rock can be forced to the surface. Over a relatively short period of time, this can cause a volcano to form. This pressure if often the result of the Earth's plates pushing against each other, which is why volcanoes are often found in areas where earthquakes appear regularly.
The core of the Earth is not its only source of heat. Radioactive decay underground creates a significant amount of heat as well, and this heat can influence the formation of volcanoes. The total heat created by this decay, however is far less that the residual heat in the core of the planet.