How do you say this right This paper will examine and explain what makes a good leader.

A better phrasing for this would be "This paper will define and examine the qualities a good leader possesses." your phrasing isn't necessarily incorrect, it's just awkward. "examine" and "explain" are just a little too close, so it sounds a little repetitive; i chose a different verb for you but you can pick something else to put in there. additionally "examine what makes a good leader" doesn't exactly make sense, so you need to rephrase it with a new verb that fits your subject.

In this paper I will explain what makes a good leader.