A little month, or ere those shoes were old with which she followed my poor father's body, like Niobe, all tears—why she, even she, married with my uncle, my father's brother, but no more like my father than I to Hercules
Act 1 Scene 1 Horatio & The Ghost
Now, sir, young Fortinbras, of unimprovèd mettle hot and full, hath in the skirts of Norway here and there shark'd up a list of lawless resolutes for food and diet to some enterprise that hath a stomach in 't; which is no other
Avenge me, son. You're apart of my blood-line and you must do what's right for me.
Act 1 Scene 5 [Ghost Talking]
I am thy father's spirit, Doomed for a certain term to walk the night and for the day confined to fast in fires. Till the foul crimes done in my days of nature are burnt and purged away.
Hamlet was very bummed, and very hurt. He was upset at Gertrude, he cannot even speak in full sentences, and that's how you can tell. he compares her to Niobe, who grieved so bitterly for her dead children that she turned to stone.
Horatio speculates that the Ghost's appearance, in full armor, on the castle battlements is related to Denmark's troubles with Norway. But he's wrong: Old Hamlet's Ghost actually returns to ask his own son to avenge his murder. It seems pretty clear that Shakespeare wants us to pay attention to father to son relationships in this play.
The Ghost is very fearful as of this moment. Shakespeare gives the Ghost the feeling of having emotions, The Ghost have forgotten they've lived in the past. This is very important to the play because the audience understands that Shakespeare is trying to be realistic.