O my dear Gertude, this, Like to a murdering piece, in many places Gives me superfluous death.

Extra death, excessive death, too much death, which, of course, there absolutely can be. The TV show I was watching last night had superfluous death. But any death that happens to YOU is too much. You don’t need extra death to have too much death as an individual. Death on top of death isn’t really possible. You really only need one to have too much of it.
But I suppose Claudius feels as though he’s being killed in multiple places. He’s getting the scattershot of the cannon, feeling lots of hits. But he is actually fine. That he can stand there to say this would suggest he’s doing okay.

Wherein necessity, of matter beggar’d, Will nothing stick our person to arraign In ear and ear.

O will people put some poison in one another’s ears? Is that a thing you’re worried people might do? Even if it’s metaphorical poison?

This is such a curious turn of phrase, though. It’s like, built to be unclear. It’s a very convoluted idea after a series of fairly concrete images. It’s abstract.

Necessity personified and going around trying people in other people’s ears due to lack of information? It’s a hard right turn in this speech. Very very interesting.

Feeds on his wonder, keeps himself in clouds, And wants not buzzers to infest his ear With pestilent speeches of his father’s death

Feeding on wonder and keeping one’s self in clouds doesn’t necessarily sound bad. It sounds rather dreamy certainly. It makes me think of Elfine in Cold Comfort Farm who periodically drifts through the woods spouting poetry. I have some sympathy for this romantic impulse for the desire to eat awe, to float, to never touch the ground.
But of course in this case, the wonder is not at the beauties and majesties of nature and the clouds are not likely the white puffy kind that angels like to play on. But it’s interesting that Claudius has chosen these words with their usually positive connotations to describe something that he definitely does not think is good. The sentence has a rather beautiful quality of starting in this light, airy place and slowly descending through buzzers to infect to pestilent and ending on death.

Last, and as much containing as all these, Her brother is in secret come from France;

This means that Claudius received this information before he walked in to this scene and is therefore not just responding to Ophelia but to the possibility that Laertes will show up and see it – or show up and depose him – or just generally make trouble. It’s a lot to hold in the background – to watch a woman go mad, all the while knowing her brother is not far away, volatile and ready to explode. I guess I have timeline questions, though. Because did Laertes start heading from France as soon as he heard about his father’s death or before? How long has this plan been in progress?
And how long did it take to get from France to Denmark back in the day? I’m guessing that’s not a SHORT journey. And so if he came the second he heard about his father and then traveled … we’re actually talking about a fairly serious gap in time. Is it a week since the arras? I guess that’s enough time for an Ophelia to go crazy.

Poor Ophelia Divided from herself and her fair judgment, Without the which we are pictures, or mere beasts:

He starts off kindly enough with this thought.

Yes, Ophelia is divided from herself and her judgment. She is disassociative and outside her own norm.
But then he kicks her while she’s down – calling her a picture and a beast. Not directly, of course – but he first says she’s divided from her judgment and adds that people without judgment are pictures or animals. Ipso facto and so on.
Pictures is funny, though. We are pictures without judgment? Pictures?! Pictures are an awfully static analogy for a person without their judgment.
A person without judgment may be many things – but still as a picture is not one of them. A beast, I can see. That analogy makes sense. Maybe what that last sentence is about is Claudius choosing the better analogy. And he settles on the same one Hamlet found for man if his chief nature be but to sleep and feed.

And we have done but greenly, In hugger-mugger to inter him;

Claudius? Is that you? Admitting to a mistake?
You’re clearly not a contemporary politician, that’s for sure. Of course, the mistake was to bury a man in secret and he calls it green, not wrong. That is, it might suggest that there’s a way to bury a man in secret with more practice and experience.
Meanwhile, I love greenly as a descriptor of behavior. If we could get all of us to use adverbs again more frequently – I vote for more of this one.
Also, hugger-mugger is a glorious word that I would like dis-interred.

The people muddied, Thick and unwholesome in their thoughts and whispers, For good Polonius’ death;

Claudius is generally really good at politics – but this sending Hamlet to England business is kind of a political mistake. With Hamlet gone, people are free to think whatever they want and they are likely to be predisposed to be on Hamlet’s side. So by sending Hamlet away, he’s got no murderer and thereby probably casts some suspicions on himself. If he’d kept Hamlet nearby, he’d have had to have some kind of public justice. Hamlet would likely not deny his murder of Polonius. I guess the danger, though, would be that in confessing to Polonius’ murder, he might reveal why he killed him, which might reveal Claudius’ actual murder.

And he most violent author of his own remove.

There’s a song that always made me laugh called “Railroad Bill”. In it, the folksinger is trying to get the subject of the song to cooperate with his wishes. He wants Railroad Bill to climb up a tree and rescue a kitten but Bill refuses. They go back and forth for a bit until a series of violent events sweep through and kill Bill. The cat comes down and has some milk. Bill is survived by a wife and two small children. I think of a violent author of his own remove like this – as if Railroad Bill sang the song about himself.

Next, your son gone.

Next up in disingenuous griefs…Claudius mourning the loss of the man he sent away to his death.
It would be funny if he forgot that Gertrude doesn’t know about the execution order for Hamlet and he started to say “your son dead” and then caught himself to say “gone.”
It’s so much fun to watch someone caught in a lie.