I have to write an essay using Aristotle's definition of a tragic hero to compare the tragic stories of Othello and Oedipus, and then after that, I have to state which is a better tragic hero than the other. How do I start off this essay, and how do I compare them?

A good place to start such an essay might begin with a definition of the term, “tragic hero,” as defined by Aristotle in Poetics, and how that particular character elicits an emotional connection in and builds a relationship with his audience. While doing this, one might also touch upon the reasons why audiences are drawn to tragedies and what they gain from indulging in an instance of fictional schadenfreude.
With the definition of tragic hero firmly in place, one might next begin an examination of the characters, Othello and Oedipus, noting how each embodies (or perhaps does not embody) the Aristotelian ideal: how do they elicit respect, fear, and pity in their audiences? What fatal flaws lead to their respective downfalls, what do they learn as a result, and lastly, how does each accept or reject their fate? A thorough examination of the character arc--the journeys that Othello and Oedipus make--as well as a discussion of these, might also prove very useful when arguing the last section of your question, that of tragic degree.
Although some scholars and critics consider the character of Oedipus the penultimate tragic hero, your experience might differ. It might be helpful to think about whom you respected or related to more at the start of each play, whose journey engaged you more, and ultimately, which ending you found more satisfying.  
 
 
http://www.classics.upenn.edu/myth/php/tragedy/index.php?page=oedhero