A slimy little weasel of a P. I., Donald Briggs, comes into the small town of Logan City and attempts to blackmail a woman, Mrs. Adams, as to the real name of the father of her grown son, Sam, who is about to marry a girl who is a member of a prominent local family. At the same time, he was actually hired by the father of said girl to find out the identity of the boy's father before his daughter marries Sam. It turns out that Sam's mother changed her family name for herself and her son after her husband and Sam's father was executed for murder18 years before.
Perry Mason is inserted into the situation when the man who originally hired Briggs, Clyde Waters, hires Mason concerning Briggs' attempt to blackmail him. It morphs into a murder case when Briggs is found dead in his hotel room from poison gas, and it is found out that Sam had gotten into a fight with Briggs AND did I mention Sam is a chemical engineering student? Sam is arrested for the murder of Briggs.
Mason had already been looking at the transcript of the murder trial of Sam's father, sees some inconsistencies, and feels that the two murders are related. At this point, early in the first season, Perry Mason is being presented as a more conservative figure than he was in the first episode, not trying to play games with evidence and not being physically threatening to his clients. Plus here, in small town California of the 1950s, Perry is presented with a much more conservative court than he deals with in Los Angeles, is considered somewhat of an interloper by the townsfolk, and thus has to adjust his courtroom style.
You might wonder why there was so much fuss over what the father of somebody did. Until the 1970s there was a widespread belief in eugenics - that what your parents did was a good predictor of the kind of person you'd be even if you never knew the parent in question.