How would you write an analytical essay about the importance of dialouge in Top/Dog UnderDog

This is a good question. There are several ways I think you could go with this, and several things you could possibly focus on. Topdog/Underdog is a play by Suzan-Lori Parks about two brothers, Lincoln and Booth (named for Abraham Lincoln and John Wilkes Booth as a joke). The way they speak to each other throughout the play is very telling of their relationship, so perhaps your paper could analyze the way that the brothers speak to each other, and what each one's individual voice says about that character and how he feels about his brother.
Another way you could come at this is perhaps to think about the fact that the two brothers were named after two historical enemies, one who was ultimately killed at the hands of the other. Does the dialogue in the story foreshadow its violent end?
You could also discuss the dialect used by the characters. It is very heavy and pronounced, so maybe you could take a look at how this dialect is used to create the characters and how it relates to their upbringing, way of life, etc. What does their dialogue and way of speaking tell us about where they came from? Does it help to develop and individualize their characters or to make them seem more realistic?
Even aside from all this, dialogue is arguably the most important part of the play, because it is just that: a play. In a play, dialogue is the thing that tells the story and moves the play forward (along with the actions of the actors on stage). So on the most basic of levels, you could also argue that without dialogue, this particular play wouldn't even really exist.