Duke; wants a well governed state without being looked upon as a tyrant.
Obstacles; state overrun with vice (especially sex), he hasn’t enforced laws in the past, therefore there is little precedent, the laws leave little room for interpretation making it difficult to let some off the hook while others are punished.
Angelo; wants to have power over people (power over the people in the state, power over Isabella, power of his decision over the law, ect…)
Obstacles; he’s too cruel, which creates fear in the state but not trust or love. To have more substantial power, like the Duke, your people must want to follow you instead of being coerced. Also, he will not be inheriting a state, he has to work his way up. Lastly, his own fallible character which leads him to speak of the righteousness of law while leaving his fiancee and seducing Isabella.
Claudio; wants everything to always be in his favor. For most of the play, we see him in some sort of requesting position. He begs not to go to jail, he wants his sister to sleep with Angelo, ect…
Obstacles; his arrest makes him unable to actually go or do anything, because he’s stuck in jail. This could be a reason for all of his “wanting via asking” because there’s not a lot else he can do in his position. Also, wanting everything to be in his favor tends to lead to unethical choices i.e. getting his girlfriend pregnant.