Like Maegor I Targaryen, Aegon doesn't physically appear in any ASOIAF media, and is only mentioned in two of the novels and in two of the Dunk and Egg novellas.
Aegon IV was the eldest son of the King's Hand, Viserys, and grew up whilst his cousins Daeron I and Baelor I ruled. Aegon's brother, Aemon, became a great knight and warrior during Daeron's invasion of Dorne, joining the Kingsguard. He was held captive after the war, until he was freed by Baelor's negotiations.
Upon becoming king, Baelor dissolved his marriage to his sister-wife Daena and imprisoned her and her two younger sisters, Rhaena and Elaena in the Maidenvault so the sight of them would not tempt him to carnal thoughts. Aegon was not under any such prohibition and developed a fondness for his cousin Daena, despite his marriage to his own sister-wife Naerys. On one occasion he helped Daena escape from the Maidenvault and they conceived a child. Daena refused to say who the father was, and was dubbed 'Daena the Defiant' for her wilfulness. In time she gave birth to a son, Daemon and Aegon noted with approval that he was growing up to be a strong warrior.
Baelor starved himself to death and Viserys only ruled for a single year before passing himself in 172 AC, so the throne passed to Aegon, Fourth of His Name. There were suspicions Aegon had poisoned his father.
Aegon IV is generally considered to be one of the worst kings in the history of Westeros, and is dubbed 'Aegon the Unworthy' in history texts. Although he kept the realm out of war, his personal life was dissolute and shocking even by Targaryen standards, and his actions at the end of his life paved the way for the then-bloodiest civil war in the history of the nation.
He was a decadent, corrupt ruler who indulged his passions and whims at every opportunity. Over the course of his life he had, quite openly, nine 'Great Mistresses', and flaunted them at court. Daena was not counted in this number, and their tryst was not openly admitted until long after he became king. Of the nine mistresses, several bore him bastard sons and daughters.
This was to the distress of his sister-wife Naerys, who was frequently miserable and, according to some, sought comfort in the arms of her other brother, Ser Aemon the Dragonknight. Aegon IV and Naerys did bear two legitimate children, Daeron II early in their marriage and Daenerys later on.
Aegon IV's cruelty and capriciousness was revealed when he caught one of his Kingsguard, Ser Terrence Toyne, abed with one of his mistresses. He had both killed immediately, along with his mistress' father Lord Bracken. Two of Toyne's brothers attempted to murder Aegon IV in vengeance, but instead killed his brother Aemon. Aegon IV had the house destroyed and its members outcast, but they would go on to trouble the realm for several generations until Ser Barristan Selmy put an end to their last descendant, Simon Toyne, almost a century later.
Aegon would often give out important positions and honors to nobles in exchange for them letting him sleep with their relatives. He gave Lord Butterwell a dragon's egg for letting him sleep with all three of their daughters, the position of Hand to Lord Bracken as their daughter bore him Aegor Rivers. When he had an affair with Melissa Blackwood, the Blackwoods and the Brackens having a terrible feud, he took some Bracken land and gave it to the Blackwoods. Melissa bore him Brynden Rivers, later known as Bloodraven.
As Aegon IV's children grew up, he became disinclined to pass the throne to his son Daeron, whom Baelor had wed to Princess Myriah Martell to shore up relationships with Dorne but had come to show signs of weakness, preferring the company of septons and maesters to warriors. Aegon IV's eldest natural-born son, Daemon, had instead grown up to be strong and charasmatic, a true warrior without compare. On Daemon's twelfth birthday, Aegon IV openly acknowledged him as his natural-born son and presented him with the Valyrian steel blade Blackfyre as a gift, shocking the Seven Kingdoms for Blackfyre, the blade of Aegon the Conqueror, was arguably the greatest and strongest symbol of Targaryen rule and was traditionally given to the heir to the throne. Some saw this as Aegon acknowledging Daemon as his heir.
In his later life, Aegon IV seemed to often praise the accomplishments of Daemon and his other warrior bastard son, Aegor Rivers, also known as Bittersteel, but held silent on his heir Daeron's achievements. Some whispered that Aegon had heard the rumours that Daeron was actually the son of Naerys and Aemon, and was planning to formally disinherit Daeron in favour of Daemon. It is believed by many that Aegon was responsible for these rumours, which began about the time he was arguing with his son about invading Dorne.
In the event Aegon did not disinherit Daeron. Aegon IV had become immensely fat in later life and fell ill without warning. He died quickly in 184 AC, but before he did so he legitimised all his bastards, including the four Great Bastards, as they were known, on his deathbed.
Despite the rumours, Daeron took the throne, styling himself King Daeron II Targaryen and his half-brother Daemon, now calling himself Daemon Blackfyre, held his peace for a time.
In early life Aegon IV was a strong, fit young man and a capable warrior, if completely outshone by his brother Aemon's martial abilities. In later life Aegon became immensely fat, and grew a beard to hide his round cheeks and multiple chins (this failed). His eyes became lost in the folds of his face and his legs had difficulty supporting his bulk. In the last few years of his life he took to wearing immense amounts of jewellery, such as rings and chains, and had a new crown forged, a substantial device surmounted by four dragons.
As a younger man Aegon IV could be charming, but as he grew older he became much angrier, prone to jealousy and terrifying rages where he was quite capable of killing or ordering people killed. Some of his mistresses could calm his temper, but the only person capable of fully calming him was his youngest bastard daughter, Shiera, whom he doted on. The ultimate example of Aegon IV's petulant rage was his legitimising of his bastards on his deathbed, which would eventually lead to a devastating loss of life and dynastic chaos that would weaken the Targaryen dynasty.