Claudius's "O, my offence is rank" soliloquy reveals that Claudius is unrepentant about having killed his brother and married his brother's wife. It also shows that Claudius is honest with himself about who is he and that he has a sense of right and wrong.
Claudius makes no pretense that he is sorry or expects forgiveness for the crimes he has committed. He is glad he killed his brother, because he likes being king. At the same time, he realizes he has committed a great sin: he doesn't try to rationalize this or come up with excuses for it. He says,
my offence is rank. It smells to heaven.
It hath the primal eldest curse upon't,
A brother's murder.
In other words, he has done the worst possible thing in the eyes of God in killing his brother. Later in the soliloquy, he says he has a "bosom [heart] black as death!" Claudius doesn't sugarcoat either the bind he is in. He would like to have it both ways: to have the crown and God's forgiveness at the same time. He knows he can't have both but feels it is impossible to give up the immoral path.
The soliloquy illustrates the theme of earthly corruption that runs through the play. Claudius realizes he is evil, and he realizes that as far as the world is concerned, he can get away with it:
In the corrupted currents of this world
Offense's gilded hand may shove by justice,
And oft 'tis seen the wicked prize itself
Buys out the law.
Money and power, he says in the lines above, can buy out justice on earth. However, his soliloquy also expresses a second theme of the play: that there is a higher justice that can't be tricked or bargained with. Like
Hamlet,
Gertrude, and the courtiers, he is trapped between the holier demands of heaven that his soul be pure and the practical concerns of the world, in which immoral behavior is rewarded.
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