Well, consider first that Paul is the only character who is black in the entire play.  Paul wants to fit into a white and racist society, and the only way he knows to do this is by means of an outright lie: that he is the son of the famous actor Sidney Poitier. 

I never knew I was black in that racist way 'til I was sixteen and came back here [to the United States]. I don't even feel black.

In a way, Paul is part of the racism by rejecting his heritage.  He doesn't "even feel black."  He simply pretends to be a member of the white world.  Unfortunately (fortunately?) for Paul, his fib is found out, and he first admits that "I am black" on the way to the police station.

Just the idea that Paul tries to "trick" the Kittredges shows that there is rampant racism.  Why would there need to be an attempt to pretend if racism didn't exist?  In reality, the story is more about Paul's desire to belong somehow and some way in society.  It is that longing that causes his tendency to lie and trick the Kittredges.