PLato's main contributions to metaphysics were mind-body dualism, the theory of forms, and the notion of a hierarchy of being. In his early dialogues, the characters search for the essences (sometimes called "idea" or "form") of shared characteristics. The typical formulation is "the X which makes all X things X", e.g. the Good (the absolute Good or good-in-itself) which makes all good things good, or the quality of "goodness" in which all good things participate. Platonic philosophy is technically termed "realist" in that it claims essences to exist independently of their instantiations. For Plato, the ideal of "goodness" or the "form of the good" is more real, because it is more permanent and more absolutely knowable, than the ever-changing phenomena of everyday life which exhibit good qualities. This distinction between forms, or noeta (things knowable by the mind) and phenomena or aistheta (things knowable by the senses) constitutes a hierarchy of being. The rational soul is capable of apprehending the noeta, the body and the irrational soul can perceive only phenomena.
Metaphysics means "above" physics. Although Plato was preceded by other metaphysical philosophers, most directly by Paramenides, I consider him to be THE classical/ancient metaphysical philosopher, meaning his influence was the greatest of the ancient world.
His metaphysics, as the first poster noted, has to do with Ideal Forms. These are the concepts which are the metaphysical ideas which provide the concept from which objects manifest or 'form' themselves. There is the concept of a chair and the real physical chair. The Ideal Form of the chair, which is defined by its function and form is the perfect chair. The actual physical chair is once removed from that Ideal Chair and a painting of the chair is twice removed from that Ideal Chair. This system of thinking is also exemplified by Plato's most famous "Allegory of the Cave."
People are chained, unable to move, facing one wall in a cave. There is a fire behind them and figures passing in front of the fire, also behind them. For them, reality is the shadows on the wall. Plato thought what would happen if a prisoner was freed and able to see that the figures using the fire were more real than the shadows. He further thought, what if the prisoner made it outside the cave to the Sun, the Ideal Reality. He answered this by saying the philosopher could get out and must go back in to try, perhaps futilely, to teach the others to break the bonds of their illusions of reality. The parallels are the Ideal Chair/Sun, physical chair/fire and figures, and painting of a chair/shadows on the wall. This was the beginning of a metaphysics which showed how reality is based on perception and that maybe Actuality, the objective Ideal, might be obtainable. These ideas of reality being based on perception and the concept of Ideal Ideas still circulate in philosophical discussions and even in science where the goal is objective observations: cue the Sun.
As a philosopher and teacher Plato brought forth questions such as "What is there?" Metaphysics deals with the ideas that object or things come about based on their existence and purpose. The most understood or general principle is to be something.
The idea behind the concept is that every object has a physical property that identifies or defines the object. For example, a ball is usually rounded (even though we know a football is oval). In this he gave us the simplicity of looking at forms and objects. Plato’s theories transfer into science and math as the constants exist in most objects.
Plato key concept in metaphysics is the idea that objects have a significant feature that defines the object. Plato was a philosopher who also looked deeply at the idea of object permanence and existence. He believed that to be something an object had to exist. To exist an object had to have some type of feature to equate its existence. It did not have to be a pphysical feature. For example; Voice has pitch and tone. It came from the idea of existence in general.
Plato also devised the theory that only certain people are of the proper deity to rule. The wise person has divine definition of reality and is therefore equipped to rule.