The relationship between Aerys II and Rhaegar worsened paralleling his growing madness and paranoia. He became suspicious and distrusting of everyone, including his own son Rhaegar, that he suspected to be plotting against him to take the throne.
According to The World of Ice & Fire, Aerys II completely lost his sanity after the events called The Defiance of Duskendale, when he was held prisoner by House Darklyn
Captivity at Duskendale had shattered whatever sanity had remained to Aerys II Targaryen. From that day forth, the king’s madness reigned unchecked, growing worse with every passing year.
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Once safely returned to King’s Landing, His Grace refused to leave the Red Keep for any cause and remained a virtual prisoner in his own castle for the next four years, during which time he grew ever more wary of those around him, Tywin Lannister in particular. His suspicions extended even to his own son and heir. Prince Rhaegar, he was convinced, had conspired with Tywin Lannister to have him slain at Duskendale. They had planned to storm the town walls so that Lord Darklyn would put him to death, opening the way for Rhaegar to mount the Iron Throne and marry Lord Tywin’s daughter.
The World of Ice & Fire, The Targaryen Kings. Aerys II chapter
Another episode that set them apart was the wedding between Rhagar and Elia Martell, that Aerys did not attend, for fear of being assassinated
Meanwhile, King Aerys was becoming ever more estranged from his own son and heir. Early in the year 279 AC, Rhaegar Targaryen, Prince of Dragonstone, was formally betrothed to Princess Elia Martell, the delicate young sister of Doran Martell, Prince of Dorne. They were wed the following year, in a lavish ceremony at the Great Sept of Baelor in King’s Landing, but Aerys II did not attend. He told the small council that he feared an attempt upon his life if he left the confines of the Red Keep, even with his Kingsguard to protect him. Nor would he allow his younger son, Viserys, to attend his brother’s wedding.
The World of Ice & Fire, The Targaryen Kings. Aerys II chapter
But probably the event that made evident to everyone their damaged relationship was the Great Tourney at Harrenhal, one of the most important events of recent Westerosi history, whose consequences shaped basically everything that happened since then. The court was already divided into factions supporting either Aerys II or Rhaegar, and Rhaegar was believed to be the real responsible for the tourney, that he used as a mean to gather like-minded nobles to deal with the issue of the madness of his worsening father.
His lordship [Lord Whent] lacked the funds to pay such munificent prizes, they argued; someone else must surely have stood behind him, someone who did not lack for gold but preferred to remain in the shadows whilst allowing the Lord of Harrenhal to claim the glory for hosting this magnificent event. We have no shred of evidence that such a “shadow host” ever existed, but the notion was widely believed at the time and remains so today.
But if indeed there was a shadow, who was he, and why did he choose to keep his role a secret? A dozen names have been put forward over the years, but only one seems truly compelling: Rhaegar Targaryen, Prince of Dragonstone.
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The prince, it is said, had no interest in the tourney as a tourney; his intent was to gather the great lords of the realm together in what amounted to an informal Great Council, in order to discuss ways and means of dealing with the madness of his father, King Aerys II, possibly by means of a regency or a forced abdication.
The World of Ice & Fire, The Fall of the Dragons. The Year of the False Spring chapter
The victory of the Tourney by Rhaegar also contributed to raise the suspicions of the King
Prince Rhaegar emerged as the ultimate victor at the end of the competition. The crown prince, who did not normally compete in tourneys, surprised all by donning his armor and defeating every foe he faced, including four knights of the Kingsguard. In the final tilt, he unhorsed Ser Barristan Selmy, generally regarded as the finest lance in all the Seven Kingdoms, to win the champion’s laurels.
The cheers of the crowd were said to be deafening, but King Aerys did not join them. Far from being proud and pleased by his heir’s skill at arms, His Grace saw it as a threat. Lords Chelsted and Staunton inflamed his suspicions further, declaring that Prince Rhaegar had entered the lists to curry favor with the commons and remind the assembled lords that he was a puissant warrior, a true heir to Aegon the Conqueror.
The World of Ice & Fire, The Fall of the Dragons. The Year of the False Spring chapter
As we know, the Tourney at Harrenhal set in motion all the event that lead to Robert's Rebellion and the end of the rule of House Targaryen, so from that point on, Aerys II and Rhaegar had to worry more about their external enemies rather than furthering their own plans about each other. In the end, their bad relationship ended with their own lives.
Summarizing, we can say that Aerys II and Rhaegar had a bad relationship that originated from the King's madness, and that kept worsening in parallel to the worsening of the monarch's conditions. The factions of the nobles that were behind both of them, and that had something to gain from the prominence of one of the other, also were a factor in making their bad relationship even more bitter and uneasy.
