Is Claudius completely evil or does he have some redeeming characteristics?
In order to answer this question you need to do a little brainstorming about Claudius's character. What do we see him say or do? How are we supposed to respond to those comments and actions? If you evaluate the totality of his character as it is presented throughout the play, then you can make a supportable determination of whether or not he has any redeeming qualities.
Here is a list of some of his most important words and actions:
First and foremost: He kills his own brother to take over the throne of Denmark and marries his former sister-in-law, the Queen Gertrude.
2. In his first speech he tells the court that he as a plans to try to negotiate a peaceful solution to the threat of Fortinbras.
3. He tries to act fatherly to Hamlet, but counsels him to put aside his mourning -- stating that all sons eventually lose their fathers and this display is "unmanly" and "shows a will opposed to heaven."
4. He sends for two of Hamlet's friends and directs them to spy on Hamlet and report back as what might further wrong with Hamlet.
5. He spies on Hamlet while he is talking with Ophelia.
6. He plans to send Hamlet away on state business to England.
7. He later amends those plans and writes a letter to England instructing them to kill Hamlet immediately.
8. He tries to pray, but admits that he has no intention of giving up the things he has gained with all of his past sinful acts.
8. He manipulates Laertes's anger away from himself and solely on to Hamlet for killing Polonius, claiming no responsibility in his letting a dangerous Hamlet run amok in the castle.
9. He comes up with multiple ways to use the poison Laertes brings.
10. He doesn't stop Gertrude from drinking the poisoned cup for fear of exposing himself.
In the above list I highlighted two things that may be considered to makeClaudius redeemable, but I don't see how those can out weigh the mighty list of despicable deeds that overwhelms the scales.
If you discount Claudius's actual acts and consider his qualities you could say that he is determined (to get what he wants when he wants it), in love (with his former sister-in-law), peace-minded (in his willingness to try diplomacy), shrewd (in manipulating others), single-minded (in his willingness to do whatever necessary to maintain his position), and smart (in seeing the truth of people), but most of his best qualities only serve to bring him towards evil, not good.