I put this in an edit to @KRyan's answer, but I'm not sure if or when that edit will be accepted, so to save time I'm writing it up here as well, along with expanding it greatly to include additional new details to make it more worthy of being an answer in its own right.
To provide a direct quote proving that Asmodeus not only is a god, but that he wanted to be one:
Asmodeus was not a god, but still the king of the Hells and craved every scrap and snippet of power he could gain. The tieflings in the world, well, their lives hung on his balance. He sought to make them all his slaves, because he could.
But a devil loves a deal, and Bryseis Kakistos offered him the chance to become greater than an archdevil, a very god if he took the chance
and then a few chapters later
"If you go back in time, it was never a secret that the king of the Hells wanted godhood," Lorcan said. He spoke without softness, without care. As if he meant for her to feel every blow. "Whether they offered the sacrifices or he demanded them isn't clear. But thirteen tieflings made a pact with Asmodeus—a mass sacrifice of fiend-born, plus their own souls and blood, for the chance to wield the powers of the Hells.
Both quotes from Brimstone Angels: Lesser Evils
Asmodeus is a god. He wasn't one for a very long time, but today he is. Originally he became one by consuming the Divine Spark of Azuth (killing that god in the process). Later Azuth came back to life from within Asmodeus, and the two split. In order to remain a god, Asmodeus was allowed to take the divine spark of an ancient god by the name of Nanna-Sin, in return for bringing Nanna-Sin back to life as a non-divine immortal.
Becoming a god (along with a few other small factors largely outside the scope of this answer) enabled Asmodeus to end the Blood War between the Nine Hells and the Abyss, a state of affairs that lasted for over a century before the war was apparently started again.
This tale tells us that divinity is something which can be concretely measured in some way. Gods have a "divine spark", and consuming that spark is one method to achieve apotheosis. There's some talk about the spark itself being weaker or stronger based on factors like worship, and how a spark can be brought into existing through a sufficient amount of worship, but ultimately possession or lack thereof of a divine spark is what defines true divinity.
The best source for this information is in Erin M. Evans' Brimstone Angels series of D&D novels, in particular the third, The Adversary, and the final, sixth, one, The Devil You Know. A brief account of parts of the story is told in the Sword Coast Adventurer's Guide 5th edition D&D sourcebook.
Something that may also interest you from that series — particularly the fifth (Ashes of the Tyrant) and sixth (The Devil You Know) books — is the world of Abeir. Abeir is Toril's twin world — Toril being the world on which most of the Forgotten Realms takes place. Abeir is a place forsaken by the gods and is instead ruled over by primordials and dragons. Divine magic does not work there, and neither does standard arcane magic like a wizard might use. Only the innate magic some creatures possess, like a primordial's elemental magic or a dragon's breath, work properly in Abeir.
