What are some specific examples of how Gertrude in William Shakespeare's Hamlet is easily manipulated (quick re-marriage) and weak (lust for Claudius)?
Your question is interesting because it assumes that Gertrude is weak and easily manipulated. While that case can be made, there is another way to think about Gertrude.
It is true that Gertrude was once very in love with her husband, at least according to Hamlet. He says:
But he adds:
and yet, within a month--
Let me not think on't--Frailty, thy name is woman!--
A little month, or ere those shoes were old
With which she follow'd my poor father's body,
Like Niobe, all tears:--why she, even she--
O, God! a beast, that wants discourse of reason,
Would have mourn'd longer--married with my uncle,
My father's brother, but no more like my father
Than I to Hercules: within a month:
Ere yet the salt of most unrighteous tears
Had left the flushing in her galled eyes,
Hamlet is disgusted with Gertrude both because she married so quickly and because she married Claudius, an inferior man to her former husband, Hamlet's father. One way to see this hasty marriage is to assume Gertrude was weak enough to be manipulated by her former brother-in-law into marrying him. Another possibility, of course, is that the two of them were involved in some kind of relationship before King Hamlet died, which would explain her sudden willingness to marry Claudius.
Gertrude also seems weak and easily manipulated when Hamlet talks to her in her chamber after the play. He scolds her for choosing such a man:
A murderer and a villain;
A slave that is not twentieth part the tithe
Of your precedent lord; a vice of kings;
A cutpurse of the empire and the rule,
That from a shelf the precious diadem stole,
And put it in his pocket!
In the face of his anger, Gertrude admits that Hamlet is right and asks what she should do now. He tells her and asks her not to tell Claudius that he is feigning madness; but as soon as she sees Claudius, she immediately tells her new husband that Hamlet is
Gertrude then kind of tattles on Hamlet and tells Claudius that Hamlet killed Polonius. This suggests that Gertrude just said what Hamlet wanted to hear, but as soon as he is gone she demonstrates more allegiance to her new husband than to her son of thirty years.
Another way to look at this scene, however, is that Gertrude really does understand what a horrible thing she has done and now wants to do what is right; her words to Claudius are still in keeping with her promise not to tell her husband that Hamlet is only "mad in craft." In essence, she continues Hamlet's charade by agreeing that Hamlet is crazy.
When Claudius vows to send Hamlet away, Gertrude does nothing to stop him; perhaps this is an act of weakness, but it might also be her way of protecting her son from Claudius after Hamlet kills Polonius.
In the first act, when we meet Gertrude and Claudius for the first time, Gertrude seems to take her cues from Claudius, agreeing with him that Hamlet should cease his excessive mourning; again Gertrude seems to be choosing Claudius over Hamlet. In the final act, however, Gertrude speaks her own mind when Claudius tells her not to drink and says, "I will, my lord; I pray you, pardon me."
Because we have virtually no antecedent action about Gertrude and Claudius, we must examine what we see and hear. The evidence is, at times, conflicting. Is she a weak woman or a co-conspirator with Claudius who sees the errors of her ways?