I can't find any refence say how Talos became a divine. It sound more like the empire just simply worship him when he died as emperor.

Maybe partly because he is dragon born, but in that case does Alessa became a divine too? This part is just not making any sense

Here's my copypast answer, enjoy.

There are 2 conflicting theories presented in-game that explains why and how Tiber Septim achieved divinity.

The **Orthodoxy** is the explanation used by the Imperials and it says that Tiber ascended because Akatosh saw his greatness and made him a God.

The **Heresy** is the explanation put forward by Ysmir Wulfharth, an old Nord King, who served Tiber Septim. The Heresy says that Talos is not one person but 3 and these are Hjalti Early-Beard (a Breto-Nord General and Dragonborn), Ysmir Wulfharth (An undead Nord King and aspect of Shor) and Zurin Arctus (An Imperial Battlemage).

Hjalti Early-Beard used Wulfharth to become Emperor of the Human provinces. Hjalti then changed his name to Tiber and made a peace deal with Vivec who gave him the Numidium.

Tiber Septim planned to use the Numidium to ascend to Godhead and use the Soul of Wulfharth instead of the Heart of Lorkhan (since Wulfharth was an aspect of Lorkhan/Shor) and asked Zurin Arctus to prepare the Numidium.

When the Numidium was ready, Tiber and Zurin planned a trap to lead Wulfharth into and they called Wulfharth to Rimmen, where the Numidium was. When he arrived, Arctus soul trapped Wulfharth but not before Wulfharth managed to shout him to death. Tiber then walked in and took the Soul Gem, which was called the Mantella, and played the event down as an attempt on his life by his Battlemage.

When Tiber Septim then put the Mantella into the Numidium and activated it he didn't ascend because he had the soul of an aspect of a god rather than a soul of God. Nevertheless, the Numidium was unstoppable and Tiber used it to defeat the Aldmeri Dominion.

So how did he ascend?

He ascended because of the Rebel/King Enantiomorph. The Enantiomorph is a way to become a God by re-enacting an event that beings more powerful than you did, this is also known as mantling or to "walk like them until they walk like you."

The event they re-enacted was Convention and the emantiomorph is called the Rebel/King because during the convention the Rebel becomes the King and the King becomes the rebel. During convention the Traitor (another important part of the Enantiomorph), Magnus, betrays the King, Lorkhan, so that the Rebel, Akatosh/Auriel, becomes the King.

In the apotheosis of Talos, Zurin is the traitor, Wulfharth is the King and Tiber is the Rebel and because of this re-enactment the 3 ascended together under the oversoul of Talos. However Tiber Septim is the strongest of the souls since he was the only one who did not die.

I hope I made both theories clear and I hope I made it obvious which theory is the correct one.

and if you need further proof, there is TES II: Daggerfall.

To be clear, OP, Talos is a Divine, but not an Aedroth.

They're related concepts; but different.