In what way did tarquins rule influence the founding of the republic

He was the last king to rule Rome and was known to be the ruler who based his reign on fear for that time; he had no mandate to rule. Tarquin the Proud was the one who heard capital cases without hearing out the voice of the Senate and ignored them all the way. He was also the one who took the condemned lands and assets for his own. Even if he was known for these negative things, he was also known to be good when it comes to foreign affairs through his foreign policy. But it was also because of his somehow focus on foreign affairs which alienated him from the people of Rome. It only took one incident involving his son, Sextus, to have that reason to overthrow the reigning ruler which later resulted to his defeat. That incident made the victors realize that there should be change in the way the land was to be ruled which they decided was through an election of officers thus the republic was born. 

he supported the seperation of church and state. he also promoted religious tolerance.