What were the steps taken to eliminate Plessy v Ferguson?

There were a number of steps that were taken to eliminate
the ruling from this case.  You might want to check your textbook to see exactly how
your book talks about these steps.

The basic strategy was
to start with lawsuits against "separate but equal" in higher education and then work
down until there was a basis for bringing lawsuits to integrate high schools and
elementary schools.

So, for example, the NAACP sued over
whether it was legal for Texas to maintain two different law schools -- one for blacks
and one for whites.  The point of these suits was to argue that it was really impossible
to have separate schools be truly equal.  If separate schools could not be equal, then
the whole doctrine of Plessy (saying that separate schools were equal and therefore okay
under the Equal Protection Clause) would be destroyed.

This
strategy culminated in the Brown case which overturned Plessy.