They mentioned a CNC in Star Trek 6 but he wasn't the Federation President, he was a Fleet Admiral. Though, that might have changed by the TNG era. I know they showed the president on DS9 but I don't remember if he was referred to as CNC or not.
An attempt to unify what we know about the "Starfleet Commander-in-Chief" position from the various movies and series:
In 2285 (ST III), a position of "Commander, Starfleet" was held by Fleet Admiral Harry Morrow. This position either still existed or existed again in 2372 (DS9: "Paradise Lost"), though we do not know who held it at that time.
In 2293 (ST VI), a position of "Starfleet Commander-in-Chief" was held by Vice Admiral William [Smilie / Toddman]. We have no other direct, on-screen references to a Starfleet officer in this position.
In at least the early 2270s (ST: TMP) and in 2372 (DS9: "Paradise Lost"), a position of "Chief of Starfleet Operations" existed. In the early 2270s, the position was held by Rear Admiral James T. Kirk, whose superior was Admiral [Heihachiro] Nogura. In 2372, the position was held by Vice Admiral Leyton, who reported directly to UFP President Jaresh-Inyo, who described himself as Leyton's "commander-in-chief."
There was also a "Chief of Staff" position at various points thoughout the history of Starfleet. In 2287 (ST V), the position was held by Fleet Admiral Robert [Bennett].
"Chief of Staff" may not have been the official designation of its position but rather a more colloquial term. I suggest the possibility that it was the same position as "Chief of Starfleet Operations," since we don't have any references to both positions existing at the same time that I can see (i.e., one person being called by one title at the same time as another person is referred to by the other title).
"Starfleet Commander-in-Chief" and "Commander, Starfleet" were likely the same position during the 2270s, 2280s, and 2290s. I would suggest that Admiral Nogura held the position in the early 2270s; that Fleet Admiral Morrow held it by 2285; and finally that Vice Admiral [Smilie / Toddman] held the position by 2293.
Perhaps at some point in the 2300s, possibly to avoid colloquial confusion, only the UFP President came to be referred to as the "commander-in-chief" of Starfleet. By 2372, it seems that the "Chief of Starfleet Operations" reported directly to the UFP President without any superior officer, so perhaps the "Commander, Starfleet" position that existed at that time was below "Chief of Starfleet Operations," which in turn was a renamed equivalent to the "Starfleet Commander-in-Chief" position of the 2200s.
While the UFP President was always legally and effectively the "commander-in-chief" of Starfleet, the highest-ranking position within Starfleet was at one time also referred to as "Commander-in-Chief." To avoid confusion, "Commander-in-Chief" was renamed in the 2300s to "Chief of Starfleet Operations," and the former "Chief of Starfleet Operations" position was renamed "Commander, Starfleet."
Chief of Staff / Chief of SF Operations -> Commander, Starfleet / Starfleet Commander-in-Chief -> UFP President (commander-in-chief)
Commander, Starfleet -> Chief of SF Operations / Chief of Staff -> UFP President (Commander-in-Chief)
(There is also, of course, the possibility that "Commander, Starfleet" in 2372 was still the same position as "Starfleet Commander-in-Chief," who had delegated Presidential direct reports and liaison responsibilities to Chief of SF Operations Vice Admiral Leyton. However, considering Leyton's deftness at which he maneuvered for his almost successful coup, I prefer the idea that Commander, Starfleet was a position (probably directly) below his and that he had convinced Commander, Starfleet to support the coup.)
Little know fact. The rank of Commodore was know to the TNG costuming department. It was on their rank emblems cheat sheet. There is a picture of it around on the Interweb thingie (I can't remember where at the moment.)