Beginning with Baron von Strucker and Kraken, the HYDRA High Council expanded itself through the decades largely from Kraken's efforts handpicking and molding certain individuals to rise in the ranks of HYDRA. He did this for Viper, the Gorgon and the Hive, who all rose in power to join the HYDRA ruling body.
The HYDRA High Council was created by Jonathan Hickman and Stefano Caselli early in the Secret Warriors series. It was a means by which the complicated and sometimes conflicting history of HYDRA's leadership could be sorted out.
Major Story Arcs
For further information: Nick Fury: Agent of Nothing
After dealing with Skrull infiltrations, Baron Strucker decided it was time to convene the High Council at HYDRA's most treasured stronghold, Gehenna. Kraken, the Hive, Viper and Madame Hydra all came, and they quickly set about resurrecting the last of their number in Japan, the Gorgon. Strucker's then revealed his purpose for gathering them. Nick Fury had learned of HYDRA's true structure and how it had been running S.H.I.E.L.D. for so many years. Rather than await Fury's response, he wanted war. The leaders of HYDRA all ultimately agreed.
Wake the Beast
For further information: Wake the Beast
The High Council was betrayed by one of its own. Madame Hydra, who turned out to be Contessa Valentina Allegro de la Fontaine, also turned out to be a sleeper agent for Leviathan, an old enemy of HYDRA's that was now reawakening thanks to Fontaine's betrayal. She shot and killed Viper in the process of stealing an alien power source for Leviathan's usage, but the Hive was able to revive Viper, mutating the woman in the process. In her new mutated state, Viper retook the mantle of Madame Hydra.
Leviathan effectively declared war on HYDRA, and the remaining High Council decided that HYDRA would respond in kind.
The fierce war with Leviathan occupied most of the High Council's attention, but Fury was not forgotten. They knew Fury was going to make an attempt at infiltrating and destroying Gehenna, so Strucker was there waiting for him along with the Gorgon, Madame Hydra and many HYDRA operatives. The Gorgon achieved what he wanted by fighting and killing Phobos, but beyond that, it was a failure for HYDRA. Fury and his Secret Warriors managed to deploy and trigger their bomb, getting away as Gehenna crumbled.
This defeat fractured the High Council. The Gorgon and Madame Hydra blamed it on Strucker's obsession with Fury and left, taking a sizable portion of HYDRA's forces with them. Kraken also held Strucker's failures against him and forcibly took over command of the remaining HYDRA force. The High Council then sparked a civil war in HYDRA between those who went with the Gorgon, namely Madame Hydra and the Hive, and those who remained loyal to Kraken.
Wheels Within Wheels
Kraken was revealed to be Jake Fury rather than the true Kraken. He had killed and replaced the original years before Strucker convened the High Council, and he had been helping his brother Nick Fury orchestrate events the entire time to effectively make HYDRA and Leviathan do most of the heavy lifting in destroying each other. In his final act as Kraken, he put Nick in position to finally execute Strucker.
This ended HYDRA's civil war with the majority of those who followed Kraken switching sides without their leader. The Gorgon and Madame Hydra were what remained of the HYDRA High Council, with the Hive's fate unclear after his last remaining base was destroyed by Mikel Fury and Team Gray of the Secret Warriors.