How much of an atoms mass is is within the core?

Most of it.

The core of an Atom is called the nucleus, it has all the atom's Protons and Neutrons in it. Neutrons are very slightly heavier than Protons and together they are most of the mass.

Outside of the nucleus there are the electrons, they have a negative charge to balance the Protons but very little mass, only about 1/2000 of the weight of a Neutron. SO in short.

Most of it.