What is the Islamic legal ruling on someone who tortured a cat and strangled it to death? Is there expiation for this sin, and how can he control his soul which incites evil?

It is forbidden to kill a cat without a legitimate Shar'i purpose. Whoever does so must repent sincerely and regret their action. No expiation (kaffarah) or ransom (fidyah) is required of them. Instead, they must work on purifying themselves by obeying Allah and associating with righteous people.