How did Plato build on the ideas of Socrates?

He was Plato's teacher. Everything Plato knew or felt about philosophy, he learned it from Socrates. Socrates' execution also deeply affected Plato. His political and moral philosophy is strongly influenced by his hatred of the system that got Socrates killed.

The Athenian philosopher Plato is one of the most important figures of the Ancient Greek world and the entire history of Western thought. In his written dialogues he conveyed and expanded on the ideas and techniques of his teacher Socrates. The Academy he founded was by some accounts the world's first university and in it he trained his greatest student, the equally influential philosopher Aristotle.