There are two principal approaches one might take to writing an essay on the career of Geoffrey Chaucer. The first is to regard him purely as a writer and examine the development of his works, probably beginning with The Book of the Duchess and ending with The Canterbury Tales. Depending on the length and detail required, you might go on to examine Chaucer's immense posthumous influence, since the career of a writer does not end with his or her death.
In the case of most Medieval writers, this would be the only way to approach the task, since we know very little about their lives. Chaucer, however, is the happy exception. Since he was a public servant, a courtier and a diplomat, we know a great deal about Chaucer's life. The balance you strike between life and literature is for you to determine. Chaucer is an exceptionally interesting subject in both respects.
A particularly fruitful approach might be to use the extensive information available about Chaucer's life to provide background for an essay about his literary career. For instance, in looking at The Book of the Duchess, you could write about the legendary Duke of Lancaster, John of Gaunt, who was Chaucer's friend and brother-in-law, despite the difference in social class. The Duchess in the title of this work is Blanche of Lancaster, the Duke's first wife. Chaucer was an impressive social climber. From a solidly middle-class background, he became very close to the royal family. His son became Speaker of the House of Commons and his granddaughter married the Duke of Suffolk. This social mobility would be an interesting background to the estates satire of The Canterbury Tales.