Plato was the oldest defender of dualism, the view that mind is a different fundamental substance than matter. While his evidence for dualism is not take very seriously anymore, he does have a few interesting points worth exploring.
I. Theory of Forms
One of the central pillars in Plato's philosophy is the theory of Forms. To illustrate this, let's use the example of a triangle. Any physical triangle, such as one drawn on a computer screen, will be less than perfect. The lines will have thickness and thus be more like really skinny rectangles. The lines will be jagged from the pixels. On a microscopic level, they will not be perfectly straight. So any physical triangle can only be a rough approximation of that perfect archetype triangle, what Plato would call the Form of Triangle. The perfect triangle cannot be physical. As a result, the physical world is just a rough approximation of the real world, which is the non-physical realm of abstract Forms. In other words, the realm of pure knowledge is the real world.
So on Plato's view, after death the soul is liberated from the human body and can have direct access to the Forms. So why does Plato think the human soul is separate from the body?
II. The Argument from Opposites
Plato says that when things become a certain way, they become that way from their opposites. If you become taller, it is because you were shorter. If you wake up, it's because you were asleep. And death comes to be only from life, and in that case then life only comes to be from death.
III. The Argument from Recollection
Plato shows how people know the Forms without having to learn about them, and so therefore must have lived before they were born. For example, people can compare two sticks and know that they are unequal, without ever having to learn what Equality is. Therefore, people already know the Form of Equality without ever having to learn about it.
IV. The Argument from Affinity
Destruction occurs in objects that are composites of many parts. The body, being composed of parts, can be destroyed. But the human mind is simple and not composed of parts. Therefore, it cannot be destroyed.
This version of dualism is not all that well supported. There are many examples of things not coming from opposites, or not returning to opposites. Age comes from youth but youth does not come from age. In addition, the arguments depend on Plato's Theory of Forms, which is contentious at best.