What does this quote mean?

Question

There's ne'er a villain dwelling in all Denmark
But he's an arrant knave.

Hamlet, Act I Scene V

My book says it means "No villain is worse than Claudius."

But this but is the same as in "It never rains but it pours."

So I thought it meant, "Every villain in Denmark is a complete villain."

Answer 1

You're right.

There's never a villain dwelling in all Denmark
But he's an arrant knave.

This is colloquial Early Modern English, and employs idioms which are not in use any more.

• There's ne'er (=‘never’) a villain means There is no villain
• but here is approximately equivalent to except, unless. When it introduces a clause headed by a pronoun, it makes that clause a negative modifier of the pronoun's referent. but he is is thus equivalent to who is not.
• An arrant knave is an out-and-out rascal.

This sentence may be paraphrased:

There is no villain living in Denmark who is not an out-and-out rascal.

Hamlet is making a savage joke: the great secret I have learned from the ghost is that all the bad guys in Denmark are bad guys. As Horatio points out, you don't need a ghost to know that.

Of course Hamlet has Claudius’ villainy in mind—“Claudius is not merely the despicable human being I find him, he is also a regicide and usurper” is approximately what is going through his mind here. But as angry and appalled as Hamlet is, he is also a Prince, trained in statecraft, and he is being very very cautious. The elder Hamlet’s revelation about Claudius is a dangerous secret which Hamlet is not going to discuss with ordinary soldiers. So he disguises the facts and his feelings in “wild and whirling words.”

Answer 2

From the context in the play, it cannot be concluded that Hamlet's line means anything specific about Claudius, although it is safe to assume that Hamlet certainly does see Claudius as an arrant knave.

I like the way catherine england puts it:

As a translation: Any villain living in denmark is an outright villain. That is, as Horatio's response points out, it's Hamlet stating the bloody obvious. Hamlet, if he's thinking clearly at all at this point, says it to avoid answering what Horatio and Marcellus are asking him, which is, what the ghost really told Hamlet. But I think Hamlet really is also just be babbling 'wild and whirling words', not coming up with clever witticisms.

He has just seen a ghost. That is not some everyday occurrence, and he is trying to make sense of it for himself, while his friends are asking him questions. So he seems to just mutter something that seems coherent.

But indeed, Horatio points out he needs no ghost to tell them such obvious things!

Answer 3

I believe none of these interpretations so far are completely correct, although they point in the right direction. The key is "but." It has TWO meanings here, both of which are relevant: but ="if not" (conditional)" and also but = "without the circumstance that, without also". If I am right, then

There's ne'er a villain dwelling in all Denmark
But he's an arrant knave.

means both

There is not a villain in all of Denmark
If he (Claudius) is not a rascal.

and also

There is not a villain in all of Denmark
that is not a rascal.

Those two interpretations parallel these two examples of 'but' in the dictionary.com link above:

Nothing would do (=be satisfactory) but that I should come in. 
    = Nothing would be satisfactory if I should not (~= cannot manage to) come in.

It never rains but it pours. = Whenever it rains, it always seems to pour (rain heavily) !

Basically, the verses you quoted are a likely case of dramatic irony. From wikipedia: "This type of irony is the device of giving the spectator an item of information that at least one of the characters in the narrative is unaware of (at least consciously), thus placing the spectator a step ahead of at least one of the characters." (< Wikipedia.)

This is dramatic irony because Hamlet is starting to make indirect innuendo about the immorality of the King and his actions, without mentioning the King himself. That innuendo will become more and more direct as the play progresses, culminating in the Murder of Gonzago spectacle, at which point Hamlet's accusations become obvious. By contrast, at this early point in the play, even if Hamlet's friends appear to have no idea what Hamlet is really talking about, Hamlet is aiming at something deep, since he just swore his friends to secrecy. That is something he would not do if he was merely making the observation that there's a lot of villainy in Denmark.