In "Fences" how is the father-son relationship depicted? How does Troy's relationship with his father compare to and shape his relationship with his son? Why do you think the play includes Lyons as well as Cory?

"Fences" is part of August Wilson's "century cycle." All of Wilson's plays deal with black life in a particular decade of the 20th-century. "Fences" examines black life in the post-World War II 1950s—1957 to be exact—and moves into the early 1960s, on the eve of the United States's intervention in Vietnam. After Troy dies, we see that Cory has entered the Marines.
The relationship between Troy and Cory is defined partly by Troy's envy of his son's opportunities and also by the fact that Troy does not know how to be a loving father. Instead, he only knows how to fulfill his paternal obligations ("I put food in your mouth and clothes on your back"), which is still more than his father, an abusive sharecropper, offered him.
The obvious of tension between Troy and Cory is that the latter demonstrates talent in football. Troy, in his youth, had also demonstrated talent for a sport: baseball. However, his ambitions were spoiled by racism. Cory, it seems, has an opportunity to pursue his ambition, but Troy's inability to let go of the past, coupled with his envy, discourages him from supporting his son. 
The title of the play is a double metaphor. "Fences" alludes to the ways in which people limit their lives, both due to circumstances out of their control (e.g., racism) and, in the case of Rose, out of love for someone else and the sense that love requires great personal sacrifice. Troy is working on the building of the fence throughout the play and does not finish it until the end of his life, around the same time that he has his final confrontation with Cory, who decides to leave home and assert his manhood by joining the Marines. Similarly, Troy left home after his confrontation with his father more than fifty years before.
Troy asserted his manhood by leaving the farm on which his family worked and to move to Mobile, Alabama, then to Pittsburgh. Northern segregation coerced him into a life of crime, which landed him in prison after Lyons was born. In his own way, Troy wants to fulfill his obligations to Cory in ways in which he was not able to fulfill his paternal obligations to Lyons. Thus, the other meaning of "fences" could also be the almost never-ending job of building and mending (as we do with fences) our relationships with those whom we love.