Left by itself in empty space a neutron star will not become a black hole. However, if a neutron star is in a situation where it can suck in matter (for example, if it wondered into a nebula or it was in a close orbit around another star)it would become a black hole if the Chandrasekhar limit is reached (about 3 times the mass of the Sun).

This limit is the maximum mass a neutron star can be before it becomes a black hole. Beyond this, the neutrons in the star cannot resist the gravitational force and collapse.

In a situation where a neutron star can pull in matter it would not be very long before it does become a black hole, as its gravity is second only to a black hole, about 70000 times the force of gravity on Earth.