Concerning Shakespeare's Hamlet, the play, and our feeling pity for Hamlet, the character, the answer to your question depends on whether you are referring to the writer or the reader. In other words, "important" to the writer or the reader?
Pity and fear are to be aroused in the audience while reading/watching a tragedy. The tragedy, in order to be effective, should arouse pity and fear. In this case, the reader should fear the manipulative powers of Claudius, and perhaps the gullibility of Gertrude, Ophelia, Rosencrantz and Guildenstern, etc. Though the play deals with royals and castle life, manipulation and gullibility can be present anywhere at anytime. The situation in Denmark throughout the play is only limited to Denmark on the surface.
The reader may also fear Hamlet's capabilities, as well. When acting spontaneously, he is capable of quick, decisive violence and cruelty.
Yet, the reader must also feel pity. Whatever Hamlet's faults, he is a man trapped in a world seemingly out of his control, at least in part. He suffers in intellectual isolation and, accurate or not, sees himself as betrayed by almost everyone around him. He acts heroically at times, and does decisively rid Denmark of its evil in the play's final scene, though at the cost of his own life.
That final scene, of course, cleanses Denmark, creating the catharsis that purges the fear and arouses pity. The conclusion should relieve the audience, make the audience feel that it, too, has been cleansed. Everything will be all right, now. The audience doesn't feel this if it doesn't pity Hamlet--at least not as intensely.
Thus, the writer must get you to pity Hamlet in order for the tragedy to be successful. And the reader must pity Hamlet in order for the play to be meaningful, and to feel the catharsis.