Claudio is the fornicating brother of Isabella. And, since fornication (you know, sex outside of marriage…) is both a sin and a capital crime in this play, Claudio is thrown in the slammer and sentenced to death. (Even though Claudio is engaged to Juliet and claims they had a "hand fasting" ceremony, it seems he had sex before he was legally married. See "Quotes: Marriage" for more on this.)
Since just about everyone else in this play is guilty of fornication (hello, Mistress Overdone and Lucio, we're talking about you), Claudio becomes a scapegoat when Angelo is put in charge of enforcing Vienna's sex laws.
Claudio's character isn't as well developed as the play's other major figures, but he's pretty central to the action of Measure for Measure and his punishment is at the center if the play's biggest question: Should sexuality (and morality in general) be legislated or monitored by the government?
Claudio is also the one character who becomes a mouthpiece for the play's concern with the mysteries surrounding death. At first, Claudio seems resigned to his fate after learning he's headed for the chopping block. But then, Claudio gets to thinking about things and decides that death is pretty scary. So scary, in fact, that he wants his sister to sleep with Angelo to save his life.
Ay, but to die, and go we know not where;
To lie in cold obstruction and to rot;
This sensible warm motion to become
A kneaded clod; and the delighted spirit
To bathe in fiery floods, or to reside
In thrilling region of thick-ribbed ice;
To be imprison'd in the viewless winds,
And blown with restless violence round about
The pendent world; or to be worse than worst
Of those that lawless and incertain thought
Imagine howling: 'tis too horrible!
The weariest and most loathed worldly life
That age, ache, penury and imprisonment
Can lay on nature is a paradise
To what we fear of death. (3.1.17)
Sound familiar? It should. Shakespeare's most famous moody teenager (that would be Hamlet) says something pretty similar in his big "to be, or not to be" speech when he calls death "the undiscover'd country from whose bourne / No traveller returns" (Hamlet, 3.1.). Literary scholar Walter Pater insisted that Claudio's speech is one of the most "eloquent" passages in all of Shakespeare. What do you think? Does it measure up to (or even surpass) the soliloquy Shakespeare wrote for Hamlet?
What's also interesting about Claudio is that, even though he defends his relationship with Juliet, he also sees his sexual appetite as something akin to gluttony and even suggests that having sex is like swallowing a bunch of rat poison:
[...] Our natures do pursue,
Like rats that ravin down their proper bane,
A thirsty evil; and when we drink we die. (1.2.3)
Yikes! If you want to know more about why one of the most natural of human desires is considered a "thirsty evil" in this play, check out what we say in "Themes: Sex."