What are the specific characteristics of a main character in a modern tragedy?

In addition to drawing conclusions regarding modern tragic protagonists from 20th century and 21st century drama, we can look to both Aristotle and Arthur Miller for help in answering this question. 
First, a short answer: Modern tragic figures, like ancient tragic figures, are notable for their plight, which is often that of the square peg trying to fit into the round hole. These characters seek a social integration that allows for individual dignity yet, for various reasons built into the character's personality/ psyche, this integration proves impossible.
The tragic figure is either unwilling to accept the place society provides for him or her or is unable to acknowledge his or her own value in the social scheme. (These two things are not necessarily different, and they are as true for Antigone as they are for John Proctor. Sometimes, the social order is linked to a larger sense of a world order, wherein essential values are questioned. Other times, the tragic figure experiences a more person need to revise or reject a broken social system.)
The tragic figure then will be (1) representative of a social group in one way or another and thus be relatable and (2) concerned with achieving individual dignity. 
Now, an elaboration on the short answer: Aristotle believed a tragic hero needed to be a person that the audience could effectively empathize with.
With this in mind, the protagonist of a tragedy should be neither too perfect nor too flawed. He or she should not be extreme or expressly exaggerated in any particular area. Thus, gods were not often the figures of Greek tragedy (and when they were, the playwright depicted them with very human qualities). 
Characters were often drawn from nobility, but they were not incredible geniuses or pure tyrants. Like Oedipus and Electra, tragic figures were characters who tended to be compromised by history in one way or another. They were capable of inspiring respect from an audience but not unthinking adulation.
These traits have carried over into contemporary drama. Modern tragic figures are also necessarily relatable and often burdened by a personal history that places them in conflict with the social and economic realities of the present.
In his 1978 article, "Tragedy and the Common Man," Arthur Miller argues that a tragic figure can be rather simply defined as "a character who is ready to lay down his life, if need be, to secure one thing—his sense of personal dignity."
The tragic protagonist is driven by "a compulsion to evaluate himself justly," whether he comes from the upper classes or has a background of a so-called common man.
Miller emphasizes the idea that a tragic figure does not need to be drawn from nobility or from aristocratic classes, but in doing so he is in no way challenging Aristotle's criteria. In fact, Miller's definition can be seen as a way of updating the Aristotelian definition so it tracks some of the changes that have taken place in Western society as the body-politic has moved away from rigidly hierarchical systems of monarchy and political oligarchy to more open systems of political democracy.
The "common man" of today is capable of inspiring audiences' empathy, just as a nobleman was in Aristotle's time. Arthur Miller thinks so, anyway, and many people (including me) would tend to agree.
Aristotle also insisted that the main character in a tragedy would achieve a moment of recognition (anagnorisis), an idea which is also echoed in Miller's definition of the tragic figure as it pertains to the achievement of a true self-knowledge.
Consider some examples from modern drama and you will see how the push toward a true evaluation of self underscores the conflicts surrounding the tragic hero. 
Willy Loman in Miller's Death of a Salesman is beset by a fear that he is a failure. He has lost step with the times and, as a result, tends to judge himself according to his past accomplishments (some of which are embellished and some of which may be entirely made-up). He struggles to find a way to see himself as a person of some dignity and value in a world where his skills are no longer needed and where his longevity at his job makes him an anachronism instead of a venerated man of experience.
The question that animates Willy's tragedy is one of individual dignity, and many people of the modern age could, can, and do empathize with the plight of a person who has outlived his profession in an time of rapid technological, economic, and social change. 
Middle-class and ambitious but falling short of his own expectations, Loman is a relatable figure who carries the weight of his past with him into the present (and suffers for it).
Similarly, Troy Maxson in August Wilson's Fences is a tragic figure, challenged to let go of his past so he can embrace his current life without bitterness. The conflicts that surround Maxson are also, essentially, related to the achievement of individual dignity, though, unlike Willy Loman, Maxson's issues are connected to race and a troubled family legacy.