In Shakespeare's Hamlet, I don't believe that Claudius ever really feels guilt. Based upon my interpretation, Claudius may know what is required of him as a king—by his subjects and society at large—but he never veers off of the course of the self-serving murderer he showed himself to be when he killed his brother, Old Hamlet.
In Act Four, scene one, Claudius seems to exhibit concern for Gertrude and for Hamlet when he inquires after his step-son. I do not see this stemming from genuine love for his wife—everything that matters to Claudius (with the exception, surprisingly, of Ophelia) is based upon keeping the crown for which he sold his soul (as the Elizabethans believed) in committing regicide. In having lost everything (a sentiment echoed by Shakespeare's tragic hero Macbeth), all he has left is the very thing that has ruined him as an ethical and moral man. When, at the end of this scene, Claudius comforts Gertrude with his plan to send Hamlet away, his intention, of course, is to have the young man murdered in England. Hamlet is acting too irrationally for the King to rest easily: is the young Dane insane or pretending? For Claudius' peace of mind, he will attempt to murder not only his stepson, but also his wife's only child. There is no guilt in this—only evil. Claudius looks at Hamlet's situation and notes:
O, come away,
My soul is full of discord and dismay (45-46).
In other words, he is agitated and fearful, but not with concern for Hamlet or Gertrude.
In scene three, Claudius puts a deceitful spin on the problem of Hamlet—who has mistakenly murdered Polonius, believing he was actually killing Claudius—to the lords gathered around him.
How dangerous is it that this man goes loose!
Yet must not we put the strong law on him:
He's loved of the distracted multitude,
Who like not in their judgment but their eyes,
And where 'tis so, th' offender's scourge is weighted,
But never the offence (2-7).
The audience learns that Hamlet is greatly loved by the people. Claudius is no fool: to take open action against Hamlet could create widespread upheaval. Old Hamlet was greatly loved and mourned; young Hamlet is the people's favorite son. Claudius' character is exposed, once more, as one concerned for his own welfare—he must guard the throne at all cost, but he must be sly to do so without splitting the kingdom between those who would be loyal to the new King, as opposed to those who might follow Hamlet into the fragmentation of Denmark and open rebellion.
When Hamlet is brought before Claudius, the young man's thinly veiled insults toward the King make disposing of Hamlet that much more desirable. Hamlet sneers in the King's face. Claudius cannot (with certainty) attribute it to madness and dismiss it. Hamlet needles his stepfather, telling him that Polonius is in heaven and if Claudius' servant cannot find him there, Claudius (in essence) should go to hell himself to find Polonius...
Where is Polonius?
In heaven. Send thither to see. If your messenger find him not there, seek him i' th' other place yourself. (IV.iii.36-37)
By the end of the scene, alone on stage, Claudius reveals his plan to have the King of England kill Hamlet, in that the other monarch owes Claudius a favor. Claudius' character is that of a scheming, murderous tyrant.
In Act Four, scene five, we have a rare glimpse into Claudius' gentler nature. It is hard to reconcile this man with one who has murdered his brother for his crown, taken his brother's widow in marriage (considered incest by Elizabethans, but a clever ploy to solidify his place on Denmark's throne), and now plans the murder of his nephew. Ophelia—with the stress of Hamlet's harsh rejection of her affections (he has been unsure who to trust) and now the brutal murder of her father—has gone insane.
When Claudius sees Ophelia's distracted and irrational behaviors, as well as her unsettling appearance, he seems deeply and genuinely affected. Unlike the man we have seen so far, he seems to express authentic concern for her. He speaks gently to her several times:
How do you, pretty lady? (43)
Pretty Ophelia— (58)
The King wants to know how long she has been suffering in such a state. When she exits, he commands Horatio to keep a close watch on Hamlet's former sweetheart.
This is the only time we see grief issuing forth from Claudius:
O, this is the poison of deep grief: it springs
All from her father's death.
O Gertrude, Gertrude,
When sorrows come, they come not single spies,
But in battalions (77-81).
In my experience, this passage is one of Shakespeare's most moving with regard to grief and loss. For an instant, one almost likes Claudius and wishes he were not so lacking in decency and upstanding character. Ophelia was never any more a part of the plan than to spy on Hamlet for Claudius.
However, within the turning of a page, Laertes arrives seeking revenge for his father's death, ready to give his soul away to avenge the person responsible for killing Polonius. Claudius encourages him, making certain the grieving son will be prepared to kill the responsible person, either "friend or foe" (155). Even now, Claudius is doing all he can to divert the blame from himself and point Laertes' wrath in Hamlet's direction. Laertes is a hot-head and a young man not well-seasoned in the intrigues of court—and he is certainly not aware of the kind of man that now sits upon the throne of Denmark. By the scene's end, with the artful twisting of the truth in Claudius' capable hands, Laertes is set upon the decimation of his father's killer, and the one who he blames for his sister's madness.
In scene seven, Claudius uses much the same excuse he had earlier in why he is unable to punish Hamlet—Claudius claims he must not break Gertrude's heart, as she dotes upon her son. He also notes how popular Hamlet is with the people. And so Claudius hatches a [seemingly] foolproof plan to draw Hamlet to his doom at the hand of the unsuspecting Laertes, a pawn for the evil sovereign.
Laertes agrees to be guided in this murderous endeavor by Claudius. Laertes promises:
To cut his throat i' th' church... (139)
By the time Act Five arrives, Claudius had tricked Laertes into believing that Hamlet, his one-time friend, is now his enemy. If Laertes' poisoned sword does not kill Hamlet, the King also has poisoned wine on hand—but stands aside, silent, as his wife drinks the wine instead and dies.
All that takes place can be traced back directly to the new King.
The catalog of acts of deceit in the play is comprehensive, beginning with the murder of Old Hamlet. In killing the king, concealing the heinous crime, and feigning love for Hamlet, Claudius initiates subsequent events that lead to further deception.
Claudius may have pity for Ophelia because of the innocence she represents in the world, something he lost sight of a long time ago. Or, he may feel grief in that she was no threat to him and therefore, not a target—though it was his actions that set the tragedy in motion, and the blame lies squarely at his feet. Ophelia is simply collateral damage, and when she loses her mind and later dies, he never misses a step. He is committed to killing everyone who will stand in his way or sacrificing anyone that can help him move his personal agenda forward. Claudius is a man completely lacking in character.