What processes does a star undergo to become a pulsar? Does it take a very specific star with a certain set of qualities such as "Just the right mass, diameter, and composition," or is it a freak accident that certain stars live out their remaining life as a pulsar?
It's generally dictated by how massive the star is. Remember what a pulsar is, it's a very rapidly rotating, highly magnetized neutron star.
Neutron stars are a category of objects which have masses between 1.4 and 3.2 solar masses. This is the end stage of stars which are not massive enough to form black holes (they're held up by neutron degeneracy pressure), but are massive enough to overcome electron degeneracy pressure (which is what prevents white dwarves from further gravitational collapse).