Who was Shabsai Tzvi and why was he considered evil?
Shabtai Tzvi was a Sefardic Jew who grew up in Turkey, became very learned, but eventually became a false Messiah. Whether he himself was bad or not is a matter of debate. In his biography/history of Shabtai Tzvi, Professor Gershon Scholem presents a very strong case that Shabtai suffered from Bi-polar disorder, a mental illness that produces extreme mood swings. It seems most likely that his "prophet", Natan of Gazza was evil and used him for his fantasies and honor.
It could be as an eminent Jewish Historian once said "Shabtai Tzvi was a Talmid Chochom who was Meshugah."
Even though he did horrible avairos, if he indeed was meshugah (mentally unstable) where is the evil in that? The fact that Natan and others helped him fool most of the Jewish Diaspora does not make Shabtai himself an evil person.
He was a false Messiah,he was Bad because mainly he said Certain mitzvos did not exist he was famous for making up the Bracha Mattir Haissurim and in the end he Converted to Islam
He was a Jew living in the 1600's who claimed to be Moshiach
To contrast with Yahu's answer.
He did many sins publicly including violating arayos (sexual sin) which is one of the 3 major sins which a person is supposed to martyr themselves for rather than transgress. Whatever his status mentally he was clearly in enough control to promote himself as the messiah even after he converted to Islam. This gigantic chillul Hashem (desecration of God's name) caused countless Jews to lose their faith and either convert or abandon Judaism.