It is easy to compare two characters—and do a good job of it—if you remember four points: * The Purposes of Comparison
These four points interrelate, but let's start with the most important: the purposes of comparison.
You can compare any two things—an apple and an aardvark, or a slug and a skyscraper. It's easy to compare things like that: a slug is ___ whereas a skyscraper is ___. You could fill in the blanks without even thinking. And that's the problem: you can do it without thinking. That's why so many papers comparing characters are (say it softly) bad and (even worse) boring. The writers don't know their purpose for comparison in general or for comparing those two characters in particular.
There are three general purposes for comparing any characters:
1) You compare things in order to find meaningful similarities and meaningful differences. The more important these are, the more important—and interesting—the comparison. That's why the whole slug/skyscraper thing falls apart.
2) As a student, you compare literary characters in order to demonstrate your understanding of the work as a whole. If you're writing about Shakespeare's Hamlet and you compare Marcellus and Gertrude, you've pretty much demonstrated you don’t understand the play well, because there's little meaningful connection between the two. On the other hand, if you compare Ophelia and Hamlet, as two adults following their respective fathers' advice to their deaths, you've demonstrated superior comprehension.
3) As a writer, you compare characters to understand the work on a deeper level. (Obviously, purpose 2 and purpose 3 are closely linked.)
And that brings us to why. Why are you comparing these two specific characters? You want to examine the two characters and the work they come from until you can complete the following statements:
"I am comparing these two characters in order to show ____...