In the ancient nation of R'lyeth, a carpenter named Yeshuah Targeryean walked into a desert to prepare for his ministry. During this time, he was visited by a tentacled monstrosity who promised him worldly power and glory if only he would submit to him. After 40 days and 40 nights, the carpenter succumbed to his pathetic human weakness and sweared fealty to the creature. On that day, the carpenter died and was reborn as: The Temujin.
The Temujin began carving a bloody path throughout the country, conquering wherever he went until he became the undisputed ruler, taking many wives and concubines along the way. However, he couldn't ensure the complete loyalty of all 300 million inhabitants. Instead of genociding all who refused to bend the knee, he decided to have them replaced slowly over many generations. He plans to have every inhabitant have a genetic connection to his bloodline, thereby ensuring that everybody living in the country would be a descendent of his. His mortal children would slowly outnumber all people who weren't related, thereby exterminating them through population growth. In some future time, everyone would be able to trace their chromosomes back to him. For this to happen, there are two problems that need to be solved here:
There needs to be a way to encourage interbreeding in order to eventually replace the entire country with The Temujin's descendents. However, a simple reward system wouldn't work, as not every child can be given a government job or some high position.
There also needs to be a minimum amount of children that need to be born each generation for this to happen, taking into account random events such as wars, famine, etc.
How can I accomplish these goals?