There is a demon named Kali on whose name Kaliyuga is there. In which of the Puranas his story is given? What is the story of the demon Kali?
Kali is stated to be the embodiment or personification of sin or paapa.
The Kalki Purana is one such Purana which has the details of his origin and details of how his family propagated etc.
In the first chapter Saunaka Rishi asks to know about the demon Kali from Suta Goswami:
Who is Kali? Where was he born? How did he become the master of the world? How does he destroy the observance of eternal religious principles?
Now, the description of how Kali came into being is as follows:
After the annihilation, the secondary creator of the universe, Lord Brahma, the grandfather of everyone, who was born on the universal lotus flower, created Sin personified, having a black complexion, from his back.
The name of Sin personified was Adharma By faithfully hearing about, cHanting, and remembering the descendents of Adharma, one quickly becomes freed from all sinful reactions.
The wife of Adharma, (Irrehgion), was named Mithya, (Falsehood) She was very beautiful, and had eyes like those of a cat They had a son named Dambha, (Pride), who was always very angry and eneigetic.
Dambha had a sister named Maya, and within her womb, he begot a son named Lobha, (Greed), and a daughter named Nikrti, (Cunning) Lobha begot a son named Krodha, (Anger), in the womb of Nikrti.
Himsa, (Envy), was Krodha’s sister. From the womb of Himsa, Krodha begot a son named Kali. Kali is always seen to be holding his genitals in his left Hand. His complexion is very black, like black ointment that has been mixed with oil.
Kali’s abdomen is like that of a crow, his face is frightening to behold, and his tongue is red and appears to be full of greed. His appearance is very fearful and a bad smell emanates from his body. Kali is very fond of playing chess, drinking wine, enjoying the company of prostitutes, and associating with gold merchants.
Kali's family lineage always propagates through incestuous relations as is stated below. Descriptions of how Kali and his descendents will behave towards achieving the goal of destroying Dharma is also found in the passage given below :
Kali’s sister was Durukti, (Harsh Speech). From Durukti’s womb, Kali begot a son named Bhaya, (Fear), and a daughter named, Mrtyu (Death). Bhaya begot a son named Niraya (Hell) from the womb of Mrtyu and Niraya begot ten thousand sons in the womb of his sister, Yatana (Excessive Pain). Thus I have described the destructive progeny of Kali, who were all blasphemers of genuine religious principles.
All these relatives of Kali were the destroyers of sacrifice, study of the Vedas,and charity, because they transgressed all the Vedic principles of religion. They were reservoirs of mental distress, disease, old age, destruction of religious principles, sorrow, lamentation and fear.
These descendents of Kali are found wandering everywhere throughout the kingdom of Kali, destroying the people of the world They are misguided by the influence of time, very restless by nature, full of lusty desires, extremely sinful, very proud, and violent even to their own father and mother Those who are known as twice-born among them are devoid of all good behavior, without any observance of proper etiquette, and always engaged in the service of sudras.
A brief account of Kali is found in Bhagavata purana 4th Canto, Chapter 8.
SB 4.8.1 — The great sage Maitreya said: The four great Kumāra sages headed by Sanaka, as well as Nārada, Ṛbhu, Haṁsa, Aruṇi and Yati, all sons of Brahmā, did not live at home, but became ūrdhva-retā, or naiṣṭhika-brahmacārīs, unadulterated celibates.
SB 4.8.2 — Another son of Lord Brahmā was Irreligion, whose wife’s name was Falsity. From their combination were born two demons named Dambha, or Bluffing, and Māyā, or Cheating. These two demons were taken by a demon named Nirṛti, who had no children.
SB 4.8.3 — Maitreya told Vidura: O great soul, from Dambha and Māyā were born Greed and Nikṛti, or Cunning. From their combination came children named Krodha [Anger] and Hiṁsā [Violence], and from their combination were born Kali and his sister Durukti (Harsh Speech).
SB 4.8.4 — O greatest of all good men, by the combination of Kali and Harsh Speech were born children named Mṛtyu (Death) and Bhīti (Fear). From the combination of Mṛtyu and Bhīti came children named Yātanā (Excessive Pain) and Niraya (Hell).