It is interesting to think about Claudius in terms of whether or not he had any regrets at all, or true feelings for Gertrude. One could almost argue that he cared for her as he tells her not to drink from the poisoned cup in Act V. But when she said, "I will, my lord; I pray you pardon me," he doesn't do anything to stop her - I mean, good grief! He's the king - he could have yelled for her to stop, screamed at the guards to stop her, even ran over to her and taken the cup from her or knocked it out of her hand. But he just lets her drink the poison. Then when she falls down, dying, Claudius makes an excuse - "She swoons to see them bleed." He didn't even try to help her! At that point, for me, I knew there was no conscience in the man. He lets his wife die and makes excuses, hoping to somehow continue this charade he's created. "Scumbag" comes to mind!
Claudius is an ambitious man who is driven solely by his desire for more. He desires to have his brother's kingdom and wife so he murders his brother and finds himself with both. He almost gets away with it, even though it was suspicious, if weren't for the ghost of the murdered King Hamlet who came seeking revenge. It is his ambitious drive, which in and of itself isn't the worst thing in the world to possess, that leads to his own death. His strength of ambition turns to a weakness because he is so overcome with it and cannot control it. He uses people to get what he wants with no regard of what might happen to them. In the end he loses all that he sought, his wife, whom he inadvertently poisons, his throne, and his life. It seems that each character in the book has some character trait which is a strength that turns to weakness and ultimately leads to his or her demise.