Absalom was David’s third son, after Amnon and Daniel. At the time of his plot to overthrown his father, Absalom was probably the most likely next king.
Amnon, the firstborn, was dead; Daniel might not have survived childhood (or he died in a later battle) since he wasn’t mentioned beyond having been born to David in Hebron (see 1 Chronicles 3:1-9).
So Absalom didn’t have to plot for the throne: he could have just waited.
On the other hand, it’s possible Absalom never had a chance after killing Amnon. Perhaps the years of exile from Israel, followed by more years of being excluding from court, led Absalom to believe that David would never willingly hand him the throne, even after death.
A four-year conspiracy to take the throne by ‘stealing the hearts’ of Israel started:
2 Samuel 15:2-4 [Absalom ] got up early every morning and went out to the gate of the city. When people brought a case to the king for judgment, Absalom would ask where in Israel they were from, and they would tell him their tribe. Then Absalom would say, ‘You’ve really got a strong case here! It’s too bad the king doesn’t have anyone to hear it. I wish I were the judge. Then everyone could bring their cases to me for judgment, and I would give them justice!’
It suggests David had lost touch with the people of Israel: why else would Absalom’s judging be so attractive? Even more to the point, how could Absalom do this for four years and not be found out?
It was only when Absalom called his supporters together and marched on Jerusalem that someone thought to tell David, and all he could do was flee with his closest supporters.