In Hamlet, would Queen Gertrude be sort of responsible for Hamlet's strong feelings against both herself and King Claudius?
Definitely. I think if you look at Hamlet's first soliloquy in Act I scene 2, after Claudius and Gertrude have left and Hamlet is left alone, you can clearly see that Hamlet's feelings of anger and antipathy towards his mother and Claudius stem at least in part from his mother's behaviour and her quick remarriage and apparent forgetting of his father. Consider what he says about her in this soliloquy:
..why, she would hang on him,
As if increase of appetite had grown
By what it fed on; and yet within a month!
Let me not think on't: Frailty, thy name is woman.
O most wicked speed, to post
With such dexterity to incestuous sheets...
Hamlet is actually very harsh in the way that he talks about his mother, but let us remember, as Hamlet repeats again and again, that there was only a month between his father's death and then Gertrude's marriage to Claudius. Perhaps then, we could argue that his denunciation of Gertrude's character and of women in general is at least partly justified. Therefore, when we consider Hamlet's words in this soliloquy, we can see that Gertrude's actions are definitely responsible for Hamlet's feelings towards both her and Claudius.