Read the play or sonnet which the character appears. Try reading it aloud with friends while you read the part of the character that you are attempting to analyze.
Discern the characters role in the story. Decide whether they are acting as the antagonist, protagonist, or whether they serve another function like soothsayer or catalyst.
Try psychologically analyzing the character. If they are a main character, then you should have adequate knowledge to ascribe motivations and fears to them. Reread some of the characters major passages in modern language to help with understanding their motivations.
Find criticism of the play where the essay seeks to analyze the characters specifically. Published criticisms of all Shakespeare's plays are numerous, so this shouldn't be difficult. Think about whether the character serves a dual literary purposes such as acting as the voice of God, or representing death or misfortune, or contrasting innocence or sin.
Pay close attention to how the other characters respond to the character under analysis. Decode Shakespearean language and look for jokes, puns or possible analogies used to describe the character. Also pay attention to atmospheric cues when the character is around, like wind blowing or moving shadows.