In act 1, scene 2, Claudius is characterized by his own words and through Hamlet's soliloquy.

Claudius uses the high, formal speech of a king as he addresses the court and various courtiers as well as Hamlet. Shakespeare characterizes Claudius as a smooth speaker, a man who uses the literary device of juxtaposition to segue into his own concerns. Claudius juxtaposes wisdom and sorrow and deftly turns from mourning for the dead king to thoughts of himself:

Yet so far hath discretion fought with nature

That we with wisest sorrow think on him

Together with remembrance of ourselves.

How is it that the clouds still hang on you?

Claudius isn't directly characterized in scene, the reader only learns of his character by thinking carefully about how he acts and what he says. In his monologue to the court and later in his monologue to Hamlet, Claudius uses some literary techniques which, by extension, characterize him.

Examples: We uses the word choice of "we" -- sometimes called the "royal we" when talking about everyone present who shares in the emotion of grief for the death of King Hamlet. Kings speak like this to show their shared humanity with their people. Here Claudius is using it to convey a shared emotion, but we learn quickly that he isn't the least bit mournful of his brother's death.

He knows that the quick marriage to Gertrude is potentially unsettling to this audience, so he is clever and uses a series of oxymorons to convey the mixed emotions of the situation. He appears to be "getting out in front of" of the situation so to speak. He specifically mentions "delight and dole" and auspicious and dropping eye."

In his talk with Hamlet his uses an allusion to Abel, "the first corse," to convey to Hamlet that everyone has to die at some point so Hamlet needs to deal with his grief better. This allusion, drawn for the Bible, suggests that Claudius is knowledgeable of religious teaching but manipulutive in that he uses it to get Hamlet to change his ways.

In Hamlet I.ii, Hamlet describes Claudius, or at least his relationship to Claudius, as being

A little more than kin and less than kind.

This line is so much fun! First, it's a joke on the similarity between the words 'kin' (a relative outside your immediate family of parents, children, and siblings), and 'kind', and on 'kind's double meaning--the way we know it, meaning nice or considerate, and an older meaning, something like 'ancestor' or 'relative'.  Claudius, according to Hamlet, is more than a distant relative, because he is now both uncle and stepfather, but he isn't very nice or considerate.

Some literary devices in this line are alliteration between kin and kind, irony (Hamlet speaks in an aside to the audience), and a double entendre.

We learn a lot about both characters from just this one line--Hamlet is clever with words, and he's angry, but more comfortable with insinuation and double entendres than a direct confrontation. Claudius, if we believe Hamlet's take on him, is unkind, a particularly faulty part of his personality when it comes to the way he treats his family, the people he should be kindest to.

I would say Claudius is sneaky.

When Claudius talks to Hamlet in Act 1, Scene 2, he uses metaphors.

Though yet of Hamlet our dear brother's death

The memory be green,  (act 1, Scene 2)

 He also mentions “bear our hearts in grief and our whole kingdom

To be contracted in one brow of woe”

 By using these flowery metaphors, he seems to be trying to prove that he is sensitive, and cares about the former king.  As we know, this is not true.  He is the one who killed the king, after all!