Please enlighten us on who was Yazeed ibn Mu`aawiyah? He was once the Caliph of Muslims? In which era did he live and was he considered righteous?
In the Name of Allah, the Most Gracious, the Most Merciful.
As-salāmu ‘alaykum wa-rahmatullāhi wa-barakātuh.
Yazid`s full name is Yazid ibn Muawiyah ibn Abu Sufyan (Radhi allahu an Muawiyah wa Abi Sufyan). His father was Muawiyah (may Allah be pleased with him) and his mother was Maysoon. His father, Muawiyah, and his grandfather, Abu Sufyan, (may Allah be pleased with them both) were blessed with the companionship of Prophet Muhammad (peace be upon him).
Yazid was born in Shawwal 25 or 26 or 27 (AH) during the reign of the third rightly-guided Caliph Uthman ibn Affan (may Allah be pleased with him) and passed away when he was approximately 40 years of age on Thursday, 14th Rabi al-Awwal 64 (AH). He is buried in Damascus and his funeral prayer was led by his son, also named, Muawaiyah.
Yazid was appointed by his father Muawiyah (may Allah be pleased with him) to be a Caliph after his demise and thus Yazid ruled after his father`s demise from Rajab 60 (AH) till his death. He is reported to have had great leadership skills and knowledge.  He was honoured in the year 49 or 50 AH to perform Hajj after being in the first army to wage a war to conquer the city of Qustuntiniyyah (Constantinople, present-day Istanbul) regarding whom Allah`s Messenger (peace be upon him) gave glad tidings.
He met and lived amongst the Sahaba and narrated Prophetic traditions too. He is described as handsome, long-haired and tall. He had fifteen sons and five daughters from different wives.
[NB: The brief life, events revolving around him, and incidents pertaining to him can be read in Al-Bidayah wa an-Nihayah, 4: 625-636, Dar al-Ma`rifah, Beirut]
However, in relation to the righteousness and innocence of Yazid bin Muawiyah (Radhi Allahu an Muawiyah), Imam Ibn Kathir (may Allah have mercy on him) states:
وقد أورد ابن عساكر أحاديث في ذم يزيد بن معاوية كلها موضوعة لا يصح شيئ منها وأجود ما ورد ما ذكرناه على ضعف أسانيده وانقطاع بعضه والله أعلم
Paraphrase: The entire collection of reports condemning Yazid bin Muawiyah (Radhi Allahu an Muawiyah) are weak or fabricated.
Mufti Rashid Ahmed Saheb (may Allah have mercy on him), in Ahsan al-Fatwa [6: 215], states:
حاصل یہ کہ یزید سے متعلق حسن ظن کے درجہ میں مطاعن سے کف لسان اور توقف ہی اسلم واحسن ہے
Paraphrase: In conclusion, to remain silent in regards to Yazid is the most suitable and best approach.
And Allah Ta’āla Knows Best
Hanif Yusuf Patel
Student Darul Iftaa
Checked and Approved by,
Mufti Ebrahim Desai.
 [Al-Bidayah wa an-Nihayah, 4: 625, Dar al-Ma`rifah, Beirut]
[Tahzib at-Tahzib, 11: 360, al-Faruq al-Hadithiyyah, Cairo; Lisan al-Mizan]
 روى المدائني أن ابن عباس وفد إلى معاوية بعد وفاة الحسن بن علي ، فدخل يزيد على ابن عباس وجلس منه مجلس المعزي ، فلما نهض يزيد من عنده قال ابن عباس : إذا ذهب بنو حرب ذهب علماء الناس
 Al-Bidayah wa an-Nihayah, 4: 625-636, Dar al-Ma`rifah, Beirut
 [Al-Bidayah wa an-Nihayah, 4: 630, Dar al-Ma`rifah, Beirut]
 [Ahsan al-Fatawa, 6: 215, H. M. Saeed Company]