She did not merely marry a second time but was also unfaithful to her husband while he was alive; this is indubitable since the Ghost states so.
On the other Hand, the way Shakespeare sees it; she was not an accomplice to her husband's murder. This is evident in the closet scene where Hamlet cries:
A bloody deed! Almost as bad, good mother
As kill a king and marry with his brother
And she replies in genuine bewilderment: 'As kill a king!' More important evidence as to this is the fact that Claudius and her never talk about the murder of the king. There is nothing to suggest that she has any notion of his secret.
As for her character, the queen was not a bad-hearted woman. She was very shallow and blunt. "She loved to be happy, like a sheep in the sun; and to do her justice, it pleased her to see others happy, like more sheep in the sun."
It sufficed her to be beautiful, surrounded by beautiful things and shiny happy people who would smile to her sitting on her throne. And why couldn't Hamlet just marry Ophelia and forget about the whole unpleasant business of his father's death and why couldn't they all just be satisfied and tranquil. She was fond of Ophelia and to a mind like hers the imperative of love was stronger than that of duty and rank. She saw no objection to their marriage if it would make her son happy.
The only chance for the queen to see the world as it is and her own deeds as they were was for her to be made unhappy. And this is exactly what Hamlet does. He points out her faults to her, his behavior afflicts her and through this affliction...