Antecedents: Vishnu – Brahma – Pulastya – Vishravas – Kumbhakarna.
The prajapati Pulastya took Havirbhu (Manini) as his wife. Their son was Vishravas. Vishravas married Ilibila and their son was Kuber. Once Vishravas looked at Kuber angrily and in order to placate his father, Kuber appointed three maids to serve him – Raka, Malini and Pushpolkada. Vishrawas sired Ravana and Kumbhakarna from Pushpolkada, Vibhishana with Malini and Khara and Shurpanakha with Raka according to Mahabharata Vanaparva Chapter 275 verses 1-8. But as per Uttararamayana , Ravana, Vibhishana and Kumbhakarna were born to Vishravas and Kaikasi. According to Agnipurana, Pushpolkada and Kaikasi are the same person.
Ravana and Kumbhakarna were the incarnations of the devas Jaya and Vijaya. Jaya and Vijaya were Vishnu’s gatekeepers. Once when a few sages including Sanaka came to visit Vishnu, they were stopped and direspected by the two. The sages cursed that they would become asuras, but they pleaded with Vishnu who said that the curse would be lifted if they were killed thrice by Vishnu. Their first incarnations were as Hiranyaksha and Hiranyakashipu. Vishnu killed Hiranyaksha in his varaha avatar and Hiranyakshipu in his narasimha avatar. Hiranyaksha was reborn as Ravana and Hiranyakashipu as Kumbhakarna. Both were killed by Vishnu in his Rama avatar. Ravana was reborn as Shishupala and Kumbhakarna as Dantavakra. Both were killed by Vishnu in his Krishna avatar. Jaya and Vijaya then returned to Vaikuntha as per Bhagavata Saptamaskanda.
Kumbhakarna spent his childhood in Vishravas’ hermitage in the Gandhamadana mountain. This was when Kuber was made the lord of the yakshas. He was also gifted many items like the pushpak. Kumbhakarna and his other brothers were envious of this. They went away into the forests and prayed to Brahma, standing on one leg, for a thousand years. Brahma didn’t appear and they stopped eating food. Khara and Shurpanakha took care of their siblings during this time. Brahma didn’t appear even after another thousand years passed. Ravana began sacrificing his ten heads into the yajna fire. Brahma finally appeared, gave him back the heads and blessed that he wouldn’t be killed by gandharvas, devas, daityas, yakshas, rakshasas, nagas, kinnaras and bhutas, and only a humna would be able to kill him. The devas were shocked and thought that if Kumbhakarna too got a similar boon, they would be in peril. They sent Saraswati to somehow twist Kumbhakarna’s boon. Saraswati began dancing on Kumbhakarna’s tongue, twisting his words and getting him a boon to sleep for six months a year. Vibhishana got the boon of remembering dharma in the face of danger and the ability to invoke the brahmastra without learning the mantra. Ravana then defeated Kuber and took over Lanka, staying there with his siblings, as per Mahabharata Vanaparva 275 and Uttararamayana.
In the war with Rama, Kumbhakarna defeated Sugriva, his ministers, Lakshmana and all the major warriors who fought for Rama. He finally fell to Rama’s arrow as per Ramayana Yudhakanda Chapter 67 verse 170. But as per Mahabharata Vanaparva Chapter 287 verse 18, he was killed by Lakshmana. But Valmiki’s version seems true since the Bhagavata Saptamaskanda confirms Kumbhakarna as an incarnation of Jayavijayas and since their release from the curse was subject to Vishnu killing them, it must have Rama who killed Kumbhakarna.
When Rama crossed the ocean into Lanka, Ravana called a meeting of his advisors to decide on the course of action. Then Kumbhakarna told Ravana that he had not abducted Sita after asking his advisors, and thus they could not be held responsible for what would happen next. But he said that since Ravana was his brother, he would do all to protect him, and that would include fighting Rama, as per Ramayana Yudhakanda Chapter 12.