The knee-jerk reaction to Claudius (for me, at least) is that he's a villain, villain, villain. Well, he is that and more.

He is courageous (Act IV, Scene 5). When Laertes barges in, after learning of his father's murder, intent on revenge. Claudius:

"What is the cause, Laertes,
That thy rebellion looks so giant-like?
Let him go, Gertrude. Do not fear our person.
There's such divinity doth hedge a king
That treason can but peep to what it would,
Acts litttle of his will. Tell me, Laertes,
Why thou art thus incensed. Let him go, Gertrude.
Speak, man."

He is human (his love for Gertrude). (Act Four, Scene 7.) When explaining to Laertes why he took no action against Hamlet even though he killed Polonius and threatened him. Claudius:

"O, for two special reasons,
Which may to you perhaps seem much unsinewed,
But yet to me they are strong. The queen his mother
Lives almost by his looks, and for myself -
My virtue or my plague, be it either which -
She's so conjunctive to my life and soul
That, as the star moves not but in his sphere,
I could not but by her."

Regretful. (Act III, Scene 3):

"Oh, my offence is rank, it smells to heaven;
It hath the primal eldest curse upon't -
A brother's murder! Pray can I not,
Though inclination be as sharp as will,
My stronger guilt defeats my strong intent,
And, like a man to double business bound,
I stand in pause where I shall first begin,
And both neglect. What if this cursed hand
Were thicker than itself with brother's blood,
Is there not rain enough in the sweet heavens
To wash it white as snow? Whereto serves mercy
But to confront the visage of offence?"

Guilty of his actions (Act III, Scene 1):

"The harlot's cheek, beautied with plast'ring art,
Is not more ugly to the thing that helps it
Than is my deed to my most painted word.
O heavy burden."

However, in my eyes, he remains a villain because, although at times he feels guilt and regret, he does nothing to make things right that he did wrong. Of course, how can you undo the murder of your brother? Confessing to the murder would be the right thing -- but doing so would lose him the crown and his queen. This he is not prepared to do.

Act III, Scene 3:

"And what's in prayer but this twofold force,
To be forestalled ere we come to fall
Or pardoned being down? Then I'll look up.
My fault is past. But oh, what form of prayer
Can serve my turn? 'Forgive me my foul murder'?
That cannot be, since I am still possessed
Of those effects for which I did the murder -
My crown, mine own ambition, and my queen.
May one be pardoned and retain th'offence?
(SNIP)
O wretched state! O bosom black as death!
O limed soul, that struggling to be free
Art more engaged! Help, angels! Make assay.
Bow, stubborn knees; and heart with strings of steel,
Be soft as sinews of the new-born babe.
All may be well. [He kneels]

While Hamlet ponders whether to kill Claudius while he is in prayer (which would send him to heaven), Claudius acknowledges his insincerity regarding his black deeds. (Act III, Scene 3): [Rising to his feet]