How much did Gertrude known about King Hamlet's murder? What evidence can you find that she might have known?Evidence of her innocence? Which is...

There is evidence that Gertrude doesn't look too deeply into things. Mostly, though she's probably just gullible and easily won over. After all, it took the Ghost of Hamlet's father to convince Hamlet that a murder had taken place, and Hamlet, himself, was plenty shocked by the news. It seems that everybody believed the story, as relayed by the Ghost that: " 'tis given out that, sleeping in mine orchard, A serpent stung me."

Indeed, during the play within the play in Act 3, Scene 2, it is only the King who recognizes the murder as a re-enactment of his own deed. Gertrude sees it merely as a play.

Then, after the play, Hamlet confronts his mother with the truth of the murder (Act 3, Scene 4), and it takes her a while to comprehend what he is saying. First she says, and it seems honestly: "Ay me, what act, That roars so loud and thunders in the index?" Then he lays it all out for her, and then there's this exchange:

QUEEN:

O Hamlet, speak no more!

Thou turn'st mine eyes into my very soul,

And there I see such black and grained spots

As will not leave their tinct.

HAMLET:

Nay, but to live

In the rank sweat of an enseamed bed,

Stew'd in corruption, honeying and making love

Over the nasty sty!

QUEEN:

O, speak to me no more!

These words like daggers enter in mine ears.

No more, sweet Hamlet!

HAMLET:

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!

QUEEN:

No more!

Even the Ghost is aware that, although she married a murderer and could have done better (or at least waited a while) she is not guilty of the knowledge of the deed or the deed itself.

Way back when he first appeared the Ghost warned Hamlet not to blame his mother (Act 1, Scene 5):

But, howsoever thou pursuest this act,

Taint not thy mind, nor let thy soul contrive

Against thy mother aught. Leave her to heaven,

And to those thorns that in her bosom lodge

To prick and sting her.

So she's guilty of incest, fickleness, dullness, vanity, and superficiality, and will, in heaven, feel those wrongs, but she didn't know there was a murder any more than anyone else did... except Claudius, the murderer himself.