In House of the Dragon, the fall of some characters has yet to be shown, but the pieces are being put into place as certain players are moved into position. If the series is anything like its Game of Thrones predecessor, when a character is taken down, it will require the actions of multiple other players, one single bad actor will not bring down the Targaryens.
In Game of Thrones, Ned Stark's fall comes only after an entire season of power plays, treachery and mistakes. It's the result of the machinations of characters who sometimes don't even know they're working against Ned or who died long before him. As Hand of the King, Ned Stark was in a very dangerous position, one that meant everyone had an opportunity to chip away at his base until he fell.
Sansa Stark
Sansa would never have done any of the things she does if she had realized what they could lead to. But she is still in small part responsible for Ned's discovery of King Joffrey's real parentage. When Eddard tells Arya and Sansa they are leaving the capital for their safety, Sansa refuses.
She says she would never leave her beloved and that they will have children with beautiful golden hair. This sparks a realization in Ned, that a brown-haired Baratheon male could never sire a golden-haired child like Joffrey. This is the first time Ned has confirmation that Joffrey may actually be a bastard.
Rhaegar Targaryen
This character may in some ways be responsible for nearly all the events that happen during the Game of Thrones series. Rhaegar Targaryen abducted Ned's sister Lyanna, starting the series of events that ended in his own death and Robert's Rebellion.
While Rhaegear died long before Ned did, and may have never even met Ned, his actions forced Robert to sit on the throne, the Lannisters get closer to power and place Ned right in the middle of it all. If Rhaegar never stole Lyanna, Ned would have lived the rest of his life in the North, unbothered by the activities in King's Landing.
Aerys II Targaryen
The Mad King, Aerys Targaryen, committed many atrocities during his reign, but one of his worst also set in motion a life for Ned Stark that would end with his head being taken in front of the Sept of Baelor. When Lyanna Stark was abducted, Ned's older brother and father went to King's Landing to protest. Aerys killed them both in a painful and horrible public execution.
He then demanded Ned Stark's and Robert Baratheon's heads. Ned had no choice but to take up arms against the crown and join in the rebellion with Robert. Aerys cursed Ned to become Warden of the North. He and his family were then given all the responsibilities that come with the power. Power that was eventually wielded against them all by the Lannisters.
Robert Baratheon
Ned Stark and Robert Baratheon were friends since childhood, both growing up in the vale with Jon Arryn the Lord of the Vale. They fought together in Robert's rebellion and their friendship throughout the show is depicted as not only effective and strategic, but loving.
So some of the blame of Ned's downfall must be put on Robert who came to Winterfell to make Ned Hand of the King. A position Ned has no interest in taking. Robert forces his hand, leaning on their friendship and Ned's devotion to duty to get him to say yes. This move brings Ned and his family to King's Landing and places them in a precarious situation between the political fighting of many strong houses.
Jon Arryn
It may be that Ned Stark got his legendary sense of duty and honor from his time spent with Jon Arryn in the Vale. Because it was Jon Arryn who first started deducing the parentage of Joffrey Baratheon. His investigation eventually was noticed by the Lannisters who conspired to kill him.
When Jon died, he left a mysterious legacy that Ned, always duty-bound, considered his responsibility to unravel. He followed Jon's footsteps to the book on the lineage of the houses in Westeros and to a base-born blacksmith named Gendry, all that would lead him to realize Jon Arryn was trying to tell Robert something. But learning this important information put a deadly target on Ned's back.
Jaime Lannister
While Jaime Lannister is not present for the downfall of Ned Stark. He sets many of the events in motion in King's Landing and is ultimately responsible for fathering King Joffrey with Cersei, the conspiracy that Ned stumbles upon. Jaime is forced to flee the city after killing Ned's household guards and stabbing the Hand of the King himself.
This attack leaves Ned Stark incredibly vulnerable and without friends in the dangerous city. He is forced to approach outside help in order to apprehend the queen for her adultery. With not enough Northmen in the city left to fight for him, Ned is forced to trust those who generally wouldn't have.
Cersei Lannister
Cersei Lannister is one of the most dangerous players in Westeros mostly on account of her ferocity and take-no-prisoners attitude. So when Ned Stark goes to confront her about her children's true father, she immediately warns him to stick his nose in her business. Ned of course is duty bound to reveal the truth and thus gets on the wrong side of the queen.
When Ned goes to confront Cersei and Joffrey in the throne room, it turns out that she has already made a deal with the soldiers Ned thought were his. The tables are turned on him, and it is all due to him warning the queen that he was coming for her, a big mistake.
Janos Slynt
When Ned finally discovers that Joffrey Baratheon is not the son of Robert and therefore not a true heir to the throne, the Hand of the King immediately begins planning on how he will safely remove the Lannister children and take control of the city. Part of his plan involves using the gold cloaks as his muscle for the challenge.
Janos Slynt is the back-stabbing captain of the City Watch known as the gold cloaks, and he apparently is up for sale to Ned. He and his army first appear in the throne room to back Ned and the small group of Northern soldiers. But with a silent nod, Janos tells his men to turn around and massacre the Northerners, then take Ned to the dungeons.
Petyr Baelish
Ned Stark and Petyr Baelish have a very tumultuous relationship that has its roots in Petyr's boyhood obsession with Ned's wife Cat. Ned does not trust Littlefinger, but the Master of Coin professes that he loves Cat so much, he would never do anything to hurt her. This of course turns out to be a lie.
Petyr steps in and offers to help Ned by buying the city guard for his use. Ned is wary but also desperate for every ally he can get. In the throne room, though, the gold cloaks turn on Ned, and its Petyr himself who puts a knife to Ned's throat, saying, "I warned you not to trust me."
Joffrey Baratheon
Joffrey Baratheon does not have very much to do with Ned's downfall until the very end when he becomes the final nail in the coffin for Ned. After the gold cloaks and Petyr's betrayal, Ned is arrested and brought before Joffrey for his punishment. It has been agreed to by every character on the show that Ned be sent to the Wall as Brother of the Night's Watch.
By this point, it is the best solution that Ned could hope for. He will return North, retain some of his honor, and even have the chance to be with his base-born son and brother. But Joffrey ignores every piece of good advice, instead having Ilyn Payne chop of Stark's head. Ned was brought low by the events in King's Landing, but his downfall would not have been total if Joffrey did not go against his advisors and order the execution.