This is a good question. If you are writing an essay on the author's purpose, argument, audience and persona, there are a number of points you need to consider.
First, you are looking at the authorial intent of the writing. In other words, you believe that you can get to the author's main purpose in writing. Many scholars do not believe that this is possible. A person who defends this view is E. D. Hirsch. In order to get at the author's purpose, close reading is important.
Second, when it comes to the author's argument, you are looking for the author's line of reasoning. In other words, you are trying to find out why the author believes in what he or she believes.
Third, when it comes to audience you are trying to find out the original audience for whom the author is writing. You want to find out the historical and cultural context.
Fourth, when it comes to persona, you are looking at how the author presents himself or herself through the work.
In the end, when writing an essay on these points, you want to find the author's intentions within the historical context. Scholars call this a grammatical / historical reading of texts.