In what ways is the Pyrrhus character similar to Hamlet and to Claudius, II.ii?
In order to really answer your question, I must tell you the background story of Pyrrhus. Pyrrhus is the son of Achilles who was killed by Paris during the Trojan war. Pyrrhus wants vengeance for Achilles' death. Since Paris is already dead, he seeks out Paris's family, including Paris's father Priam.
Pyrrhus is a foil to Hamlet: He is a son seeking vengeance for his father's death. A foil is a character whose qualities are opposite of a principal character and who brings into the foreground the qualities of that character.
Let's look toward Hamlet's speech for further understanding of the comparison:
HAMLET. The rugged Pyrrhus, like th' Hyrcanian beast,--
it is not so:-- it begins with Pyrrhus:--
The rugged Pyrrhus,--he whose sable arms,
Black as his purpose,did the night resemble
When he lay couched in the ominous horse,--
Hath now this dread and black complexion smear'd
With heraldry more dismal; head to foot
Now is be total gules; horridly trick'd
With blood of fathers, mothers, daughters, sons,
Bak'd and impasted with the parching streets,
That lend a tyrannous and a damned light
To their vile murders: roasted in wrath and fire,
And thus o'ersized with coagulate gore,
With eyes like carbuncles, the hellish Pyrrhus
Old grandsire Priam seeks.
So, proceed you. (Hamlet, II.ii)
This initial description of Pyrrhus is less than favorable than Hamlet's characterization. Pyrrhus is like a madman, covered from head to foot with the dried blood of his enemies, and not just of his father's killer, but of everyone--children, women--everyone related to Paris. He is in a crazed blood frenzy over his need for vengeance. Hamlet is the opposite: He seeks proof. He acts in measured ways. He believes he is punished by and when he slays an innocent man in the murder of Old Hamlet. He must be driven to act. He attempts to target only one man.
PLAYER. Anon he finds him,
Striking too short at Greeks: his antique sword,
Rebellious to his arm, lies where it falls,
Repugnant to command: unequal match'd,
Pyrrhus at Priam drives; in rage strikes wide;
But with the whiff and wind of his fell sword
The unnerved father falls. Then senseless Ilium,
Seeming to feel this blow, with flaming top
Stoops to his base; and with a hideous crash
Takes prisoner Pyrrhus' ear: for lo! his sword,
Which was declining on the milky head
Of reverend Priam, seem'd i' the air to stick:
So, as a painted tyrant, Pyrrhus stood;
And, like a neutral to his will and matter,
This is interesting: Pyrrhus has his sword poised to kill Priam (whom, coincidentally, his own father, Achilles, had once spared) when, for some reason, he hesitates; he is frozen for a moment. This hesitation could be a symbol of Hamlet's own hesitation in killing Claudius and seeking the Ghost's revenge for the murder of Hamlet's father, Old Hamlet.
PLAYER. But as we often see, against some storm,
A silence in the heavens, the rack stand still,
The bold winds speechless, and the orb below
As hush as death, anon the dreadful thunder
Doth rend the region; so, after Pyrrhus' pause,
A roused vengeance sets him new a-work;
And never did the Cyclops' hammers fall
On Mars's armour, forg'd for proof eterne,
With less remorse than Pyrrhus' bleeding sword
Now falls on Priam.--
Eventually Pyrrhus acts on his desire for vengeance. His excess shows that Pyrrhus is a vicious sort. He is far less noble than Achilles who could show mercy, and far less noble than Hamlet; Hamlet is quintessentially noble and merciful, though he is driven outside himself by the Ghost's demands (V.ii). In these scenes, Pyrrhus reminds the reader more of Claudius who shows no mercy: He kills his own brother for the crown and queen; he plots Hamlet's death when Hamlet seems a threat to him.
This mercilessness and the gore-covered Pyrrhus could also be foreshadowing of the ultimate carnage that will result from Hamlet's efforts at attaining vengeance.