In my opinion, Gertrude cares more about herself than her son. Whether or not she and Claudius were having an affair, she rushes into a marriage with him. This devastates her son. She is not thinking about what this will do to Hamlet, or perhaps, she is knowing that Hamlet will be destroyed by the knowledge emotionally and mentally.
Hamlet feels that she did not mourn his father's death long enough. In fact, Gertrude does say that she feels Hamlet's mindset is due to her "o'rehasty marriage" (II, 2, 56-57).
In a sense, Gertrude does care enough about Hamlet. Perhaps, she did not think it through before rushing into a marriage with Claudius.
Later on, she appears concerned with her son's mental condition or madness. She desires to find the cause. Of course, she has stated that she may be the cause.
As Hamlet is dying, Gertrude reveals her love for Hamlet. She is poisoned herself and she does not repent for any sins that she may have been hiding. One could infer that she had no part in the murder of King Hamlet.
Then again, King Hamlet, who would know her better than anyone, indicates that Gertrude is not what she seems when his ghost visits his son. He states that she is a "seeming-virtuous queen" (I.V.42-5)
Everyone knows that looks can be deceiving and Hamlet certainly has lost respect for his mother.
By rushing into a marriage with Claudius, two things could be revealed. Either she has been having an affair or she is trying to save her position as queen. What other motives could she have for marrying a man who is not only capable but has poisoned her husband?