I am not sure whether you are looking for general thoughts on this question or looking for more "mechanical" advice on how to write an analytical paper. So I will address both, the second briefly.
In order to analyze the affect of an author's background on his or her writing, you need to first research the author's life. When and where was the author born? Where did the author live? When did the author die? What significant events took place in his or her life? Was the author married? Did he or she have children? What influence might the parents or siblings have had on the author?
Once you have gathered this information, it is time to find the possible connections between the author's life and the literature he or she has written. Has the author placed the story in a setting that is familiar? Does the author write about family life in a way that suggests some of the plot might be taken from his or her own family life? Are there historical events in the story that really did take place in the author's lifetime? Is the protagonist of a story a writer like the author, or a car salesman like the author's father? These are the kinds of connections you are looking for.
Sometimes, you can find interviews with the author available and these will often include a discussion from the author of how his or her works relate to his or her life. For example, Anne Rice, the author of Interview with a Vampire, has stated in interviews that she wrote the character of Claudia, the vampire child, to immortalize her own daughter, who had died at a young age of leukemia. Authors today are often quite forthcoming about the connections between their lives and their books.
From a more mechanical point of view, writing an essay like this is much like writing any other sort of literary analysis. Find the connections, state them in a thesis statement at the end of your introduction, and then order your points, paragraph by paragraph, wrapping it all up with a conclusion. Here is an example of a thesis statement, based on a possible essay on Anne Rice:
The books of Anne Rice clearly reflect her life in New Orleans, her great interest in the supernatural, and her own family experiences.
From that thesis statement, I would write three body paragraphs, one on each point. In each paragraph, I would offer my facts about Anne Rice and then offer examples from her books, explaining the connection between them for the reader.
In my conclusion, I would remind the reader what my thesis is, that there is a connection between Anne Rice's life and her books, and I would review for my reader what points I made to support that thesis.
Be sure to provide in-text citations to document your sources for all information and ideas, as well as a works cited page at the end.