How is Hamlet a coward? Please provide evidence.

Hamlet seems somewhat cowardly because of how long it takes him to act on his father's charge that he avenge his death and destroy his murderer, the old king's brother and Hamlet's uncle, Claudius.  Old King Hamlet's ghost assigns him this task in Act 1, Scene 5, when he says, "Revenge [my] foul and most unnatural murder" (1.5.31).  However, Hamlet really doesn't do anything, besides act crazy, until Act 3, Scene 2, when he assigns an acting troupe to perform a play that is very similar to the scene that transpired when Claudius killed his father and then married Hamlet's mother after his father's death.  He even asks himself at the end of Act 2, "Am I a coward?" (2.2.598).  He wonders this because he has just seen an actor pretend to do more as part of a play than Hamlet has done as a part of his real life.  He wonders further if he is "pigeon-livered and lack[s] gall," for -- if not -- why has he not "fatted all the region kites / With this slave's offal" (2.2.604, 606-607).  If he were not a coward, he says, he would have already killed Claudius and fed his remains to the birds.  He feels "Prompted to [his] revenge by heaven and hell" and yet he has spent more time thinking and talking than actually doing anything.

Further, when Hamlet comes face to face with Fortinbras's army, he sees an example of a son who has behaved as a loyal son should; the comparison does not favor Hamlet.  He calls his revenge "dull" because it has accomplished nothing yet, and again he calls himself a "coward" (4.4.35, 46).  He goes on to say, "I do not know / Why yet I live to say 'This thing's to do'" (4.4.46-47).  Even Hamlet himself doesn't know why it has taken so long for him to act on his father's demand.  He has "cause, and will, and strength, and means / To do 't" (4.4.48-49).  Nothing stands in his way: he has motivation, opportunity, and capability, and yet, the task goes undone and he continues to think and think about it without actually taking any action.