In Shakespeare's Hamlet, what does the failed attempt that Hamlet makes to kill Claudius reveal about his nature?

It is pretty obvious that Samuel Taylor Coleridge was right when he said that Hamlet thinks too much. Shakespeare doesn't say so, but it would seem that he is obliquely criticizing Hamlet's long years at Wittenberg for creating his inhibitions about acting. According to C. G. Jung in his book Psychological Types:

Naturally only those [conscious] functions can appear as auxiliary whose nature is not opposed to the dominant [conscious] function. For instance, feeling can never act as the secondary function alongside thinking, because it is by its very nature too strongly opposed to thinking. Thinking if it is to be real thinking and true to its own principle, must rigorously exclude feeling.

Most people would agree that Hamlet's dominant conscious function is thinking. This is demonstrated in his many soliloquies throughout the play. If his dominant conscious function must "rigorously exclude feeling," then it would be impossible for him to assassinate Claudius in cold blood. He has to be experiencing strong emotions when he becomes a man of action. This happens when he kills Polonius. He is already in a highly emotional state when he is confronting his mother, and her cries for help, echoed by those of Polonius behind the arras, add other feelings--alarm, suspicion, confusion--to those Hamlet is already experiencing. He thinks he has walked into a trap. He doesn't understand why his mother is calling for the guards, or who that man is behind him.

It is easy enough to understand why Hamlet can't kill another man without being impelled by strong emotion. What Hamlet's failure to kill Claudius in Act III, Scene 3 tells us about his character is, once again, that he is an introvert who is typically absorbed in his own thoughts and finds it hard to cope with reality. That seems to be what the play is all about. A reclusive, scholarly man is thrust into the real world and has to learn to cope with manifold real-world characters and the problems they create. 

The scene in which Hamlet considers killing Claudius and then changes his mind also reminds the audience that Hamlet is wearing a sword. Shakespeare does not direct the actor to draw his sword, but the dialogue obviously calls for it.

Now might I do it pat, now he is praying;
And now I'll do't. 

This is where he draws his sword. It should have a strong emotional effect on the audience--but Shakespeare is teasing them. This is, after all, only Act III. If Shakespeare had Hamlet kill Claudius now, it would become an entirely different play. Hamlet would look like a mere assassin. He might find it impossible to claim the throne. He might find himself in deep trouble. Hamlet may not be concerned about such things himself--but Shakespeare is! 

So Hamlet continues to stand over Claudius with sword drawn until he acts in accordance with his own dialogue and sheathes his it.

Up, sword, and know thou a more horrid hent.
When he is drunk asleep; or in his rage;
Or in the incestuous pleasure of his bed;
At game, a-swearing, or about some act
That has no relish of salvation in't
Then trip him, that his heels may kick at heaven,
And that his soul may be as damn'd and black
As hell, whereto it goes. My mother stays.
This physic but prolongs thy sickly days.

But he will soon use that sword to kill Polonius and frighten his mother half out of her wits when he withdraws the blood-smeared blade and she fears he may use it on her next! It is significant that, although Hamlet did not kill the King, he thought he was killing the King in his mother's chamber. Hamlet is becoming a man of action and will continue to be more proactive and less introspective throughout the remainder of the play.