Can you tell me exactly what happened between Mu'awiya, Yazid, and the Ahl al-Bayt, especially since the questioner feels hatred towards Mu'awiya, believing that he was responsible for killing the Ahl al-Bayt, and wonders how he justified being a king without them? And does the ruling apply to them that applies to those who do good deeds with corrupt intentions, as mentioned in the Prophetic Hadith?

You have erred in your judgment of Mu'awiyah. He is a great Companion. You have contradicted yourself by lumping him together with Yazid in your judgment, for Mu'awiyah did not kill anyone from the Household of the Prophet, unlike Yazid. Moreover, leading Sunni scholars have attested to Mu'awiyah's virtue and justice, considering him among the jurists and scribes of revelation, and that he is superior to many of the Tabi'in (successors of the Companions). His fighting against Ali was not for kingship, but rather to demand retribution for Uthman's blood. He conformed to Islamic law, which dictates that allegiance is not obligatory until the caliphate is firmly established. The Companions unanimously agreed on his rule. Furthermore, Al-Hasan, who is from the Household of the Prophet, abdicated the caliphate to him, thereby fulfilling the Prophet's prophecy, peace and blessings be upon him, and Allah united two great factions of Muslims through him. Therefore, you should reconsider your judgment and repent.