My preferred take is that Graz'zt's father was Asmodeus. Of course, I also like the 4E take on Asmodeus best, where he was an angel who guarded Tharizdun's prison until Pazuzu convinced him to rebel against his god and take a shard of the Heart of the Abyss to kill him with. Perhaps Pazuzu introduced Asmodeus to Pale Night, a fellow obyrith, during this time. Asmodeus took young Graz'zt with him with the intention of one day sending Graz'zt back to his plane of birth as an archdevil conqueror.
As it stands now, Graz'zt has supposedly rebelled and embraced being a demon lord, combining multiple layers of the Abyss in a way similar to how the Nine Hells are arranged. In 3.5's Expedition to the Demonweb Pits he also attempted to establish a council of demon lords.
However, his son Rule-of-Three is said to be aiding Graz'zt in a plan to unite the various fiends. Plus, as of 5E Asmodeus was the one who commissioned the creation of the yugoloths, a neutral evil race between demons and devils (man, yugoloths sure did receive a lore downgrade; they went from the original fiends to just a project Asmodeus paid some night hags to produce). Finally, Graz'zt's on-again-off-again lover, Iggwilv, is the author of the Demonomicon and is said to know the true names of multiple demon lords (one possible ending to 3.5's Savage Tide adventure path even has Iggwilv obtaining a portion of Demogorgon's essence; 4E's Demonomicon states that this did indeed happen and that she still possesses a fragment of the Prince of Demons).
Perhaps Graz'zt, Iggwilv, and Asmodeus (and maybe even Pazuzu and Pale Night) have some kind of grand long-term scheme in mind?