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Divine Rule refers to the method by which some civilizations choose the person who would lead them. Generally, one person claims to be descended from deities and has "evidence" of divine birthright to the kingdom. For example, the Pharaohs in Egypt were "selected" by their deities, and as such, were "fated" to rule over the entire populace. Due to their "divinity," the Pharaohs were then worshiped and praised as deities themselves.
In regards to Death Note, Light Yagami believes that he is the only human that could have purged the world of evil. He has the idea that he must have been destined to find the Death Note that Ryuk dropped in the beginning of the series. That is part of the reason that he took it upon himself to become God of the New World that he would create. With the Death Note, he could now perform Divine Justice by passing judgment on the wicked. In his hands, he had the means, and in his heart, he had the will to do what no other human could do: kill every wicked and evil person on Earth.
The rest of the world was definitely willing to follow Kira and even to worship him as their new God. To assume Divine Rule, one must have the support of the deities as well as the religion's followers. Light came close to gaining power over the planet, and he would have been the first human to control all of Earth. It is in this way that he would have been God for the rest of his life. Interestingly, producing an heir before death would have secured his power as Kira: L had groomed Mello and Near, and as such prepared both to become the true heir to L. Light most likely never thought of having children or successors, as he believed he was God and would never die.