How can I start writing my compare and contrast essay between me and someone else (character)?

If I understand your question correctly, you are to write a comparison/contrast essay on a character from literature and yourself. 

Have you compiled your list of similarities and differences in your prewriting stage? Look at these and decide what commonalities you share with the character while yet differing.

Begin with some opening remarks of your own such as How often real people seem to have stepped from a novel's page,or a quote from the literary work that applies or another relative quote from an author, philosopher, etc.  Then, write a focused statement about your topic, your thesis statement.  This thesis should contain 3 main aspects of character that you are going to compare/contrast.

Be sure to make comparisons and contrasts between both persons as you write.  (Do not write about one at a time.)

Here's an example of an opener: Joseph Conrad wrote a short story entitled, "The Secret Sharer" about a man's alter-ego.  Now, it seems that I have found my "secret sharer."  For, while I have always complimented myself on my individuality, after reading _______, I find that while I differ somewhat from _____(the character), he/she and I share some striking similarities.