There are too many different approaches and methods of research that are applicable when it comes to comparative literature. In light of this, I will mention a few of them.
First, when you compare literature, you can always go down a historical route. You can compare the historical context of each literature and come to very interesting points. For example, if you compare a Greek play and its view on war and compare this to a modern play on war, there will be some interesting points.
Second, you can use various literary theories to compare literature as well. For example, you can use feminist theories to study two or more pieces of literature. Or you can use theories on power and apply them to various text. The interactions between these texts and theory can produce new questions and ways of reading.
Third, one of my favorite things to do is to study intertextuality. In other words, you can see and explore how one text uses the material from another text.