Points on how Claudius in Hamlet is an unscrupulous character but how he also has a conscience?Just short bullet points would be appreciated.
Clearly the most obvious example of Claudius's unscrupulous nature is the fact that he poisoned his brother so that he could take the throne of Denmark. Most of his subsequent bad actions are connected to that first act. Here are few other unscrupulous acts:
1. marrying Gertrude, his former sister-in-law. This marriage is considered incestuous by most everyone.
2. bringing in Hamlet's friends, Rosencrantz and Guildenstern to spy on Hamlet and report back to him about how Hamlet is feeling about things.
3. spying on Hamlet for himself, with Polonius, when they set up Ophelia to have a talk with Hamlet.
4. planning to send Hamlet away to England. At first he is trying to eliminate the negative shadow Hamlet's crazy behavior casts on the court of Denmark.
5. Later, once he realizes that Hamlet knows the whole truth of his murder of King Hamlet, Claudius not only sends Hamlet to England, but sends a command to have Hamlet killed upon his arrival there.
6. plotting with Laertes to lure Hamlet into a fencing match where the two of them will have poisoned a sword and a cup of wine in order to ensure Hamlet's death.
7. knowing full well that the cup is poisoned, he tells Gertrude not to drink, but he doesn't take any action to stop her or say any more. He doesn't want to reveal his guilt in this plot.
There are two places where Claudius acknowledges his guilt and that he has a conscience, but the awareness doesn't ever affect his actions. He doesn't take any action to correct his wrong-doings or even to own up to them. Early in Act 3, Polonius is suggesting guilty people sometimes cover up their guilt with "pious action" and Claudius remarks to himself, "How smart a lash that speech doth give my conscience . . . Is [it] not more ugly to the thing that helps it / Than is my deed to my most painted word, / O heavy burden."
Later in a Act 3, after Claudius sees the play-within-a-play and knows that Hamlet knows the truth, he is moved to try to pray for forgiveness, but in the process of trying to pray he knows that, deep down, he can't be forgiven of his crimes "since I am still possessed / Of those effects for which I did the murder -- / My crown, mine own ambition, and my queen." He is not willing to give any of them up in the name of conscience and realizes by the end of the scene that "My words fly up, my thoughts remain below [on earthly matters]; Words without thoughts never to heaven go." He may have a conscience, but it doesn't change anything for Claudius.
The key answer to your question is when Claudius prays about his acts and how he is trying to be sorry, but he can't be truly sorry because he still "possess'd of those effects for which I did the murder, my crown, mine own ambition and my queen" (III.iii.53-55). This speech is key to your understanding of his dilemma. There are also points at which he realizes that he has done wrong and that he is probably going to have to pay for what he's done. A key here is directly before Hamlet's famous "To be or not to be" speech at the beginning of Act III. He says:
How smart a lash that speech doth give my conscience!
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:
Oh heavy burthen!
The "burthen" or burden he speaks of is the murder of his brother.