In effect, did the Confederacy function as a single unit (one leader, unified laws, etc) or did it simply represent an informal cooperation between various tribes.
The Haudenosaunee certainly wasn't (or rather "isn't", since it persists in the form of the Six Nations of Grand River - I'll mainly be focused on its state prior to the American Revolution however) an informal alliance. The Gayanashagowa, the Iroquoian constitution, establishes the nature of the confederacy's political system on the confederacy level. In this post, I discussed the basics of how the men's Grand Council functioned (we unfortunately know far too little about the women's Clan Mother Council). The closest thing the confederacy has to a single leader is the Tadodaho, the member of the Onondaga delegation that chairs the Grand Council, and / or his counterpart among the Clan Mothers - the Jigonsaseh, a Seneca. But they certainly weren't kings and queens, just first among equals with limited power to influence their respective council's decisions. The Grand Council's decisions had to be unanimous, but the member nations were only obligated to not interfere with the decision rather than actively pursue it. For example, the Seneca might propose a declaration of war against one of the confederacy's enemies. If the motion passed, the Mohawks weren't obligated to join the war but couldn't do anything to undermine the Seneca's war efforts. However, the Grand Council could prevent the Seneca's declaration of war if peace if such a war would undermine other goals that the Council had agreed upon.
While the Five Nations enjoyed considerable autonomy, the Grand Council did exercise stronger control over various nations that were subjects of the Haudenosaunee. At the time of the infamous Walking Purchase in 1737, the Delaware were under the Grand Council's authority. They appealed to the Grand Council to overturn the duplicitous sale. The Grand Council reply was that they hadn't given the Delaware permission to sell land to begin with, and since they went behind the Grand Council's back on that one they got what was coming to them. Instead of arguing against the Walking Purchase, they sided with the Pennsylvanians and gave the Delaware new land where they could be better managed by the council.
This is where the rather confusing "half-kings" enter the picture. We don't know their Iroquoian titles, or exactly what their role in Iroquoian society was. It seems they filled a short-lived political niche in the early to mid-1700s. It seems the half-kings were appointed by the Grand Council to be the official intermediaries between the Council and their subject peoples. Scarouady, an Oneida, was the half-king for the Delaware, while Tanacharison, a Seneca, was half-king for the Shawnee and Mingo (a mix of Seneca, Cayuga, and other Iroquoian settlers along the Ohio River). Both operated out of Shenango (Logstown), a town along the Ohio River a little northwest of modern Pittsburgh. How much autonomy the half-kings could exercise is uncertain. While they made occasional references to the Grand Council's authority over them, they (Tanacharison especially) seemed to have acted without their approval. Whether Tanacharison's actions were at the behest of local political interests, his personal agenda, orders from the Grand Council that allowed them to act against the French while maintain plausible deniability, or a confluence of all three is unclear. Regardless, following the French and Indian War, the populations formerly under the half-kings' "authority" enjoyed considerably more autonomy and the half-king position seems to have been dissolved.