The text in question here is A Soldier's Play by Charles Fuller, which later became a movie called A Soldier's Story. Your first step in writing a character analysis of the antagonist is to figure out which character is the antagonist. Some might argue that the antagonist is Captain Charles Taylor, who doesn't want Captain Richard Davenport to investigate the murder of Vernon C. Waters. Indeed, Taylor doesn't believe that a Black man like Davenport can get any results, for Taylor is sure that white men committed the murder and, therefore, that the accusation of a Black man against them will never stick.
However, a better choice for an antagonist might be Vernon C. Waters himself. This might seem like an odd choice, because when the play opens, Waters has already been murdered. Waters, however, does appear in flashback scenes, and we can see that he is quite the opposite of the balanced and organized Davenport.
Waters has a particular vision of what a Black man should be like, and he torments anyone who doesn't live up to his standards, including C. J. Memphis, who ends up killing himself. Waters is despised by his men and disliked by everyone in general. He does feel guilt after Memphis's death, but he turns to drinking instead of solving his problems. His smart mouth gets him beaten by two white soldiers, but Waters is actually killed by two of his own men in revenge for Memphis's death.
After you choose which of these men you think is the antagonist of the play, you will need to re-read the work, paying close attention to how the character is portrayed and how he interacts with other characters. If Taylor is your choice, you could focus on his prejudices and his presuppositions. He has already made up his mind about Waters's murder, and he doesn't think that Davenport can do anything helpful. If you choose Waters, notice how he berates people who don't agree with him, and pay special attention to his ideas about the proper position of Black men and to how he treats Memphis.