The events of the cup d'etat are different than the events related to the Great Holy Knight murder. The cup d'état just happened some days before Ali finds Meliodas in the Board hat, the events of the first chapter. The murder of the Great Holly Knight happened 10 years before to that moment. Just in case you are mixing them.
In fact, there has been no much speculation about the cup d'état. The reasons seems to be related to
the position of the Former King of Lyonesse towards the prophecy of the Holy War. The captured King and all the knights knew there was going to be a Holy War, according to the prophecy. To avoid it, the king ordered the Knights to lay down their swords. Some of them didn't approve it and schemed the cup d’état. After capturing the King, they have been preparing to the war. Now the preparations are of two kinds; some of them (Hendricksen and Helbram among them) started to create new powerful Knights out of the apprentices and using the Demon blood. The others don't know about it (Dreyfus, Gildmore, Hauser).
Now, one interesting fact of all this political event is that nobody, nor the king, neither the Knights seems to know against
who they are going to fight this holy war of the prophecy. All of them have blindly believed that prophecy, even ignoring who is the enemy. But it's possible someone among them knows much more about that prophecy than what has been shown.