Prospero is often viewed as similar in nature to a playwright in The Tempest. Can you give three ways that he exemplifies this role?

There are several ways in which the character of Prospero in The Tempest by William Shakespeare functions in a manner analogous to the playwright himself.

First, Prospero at some points functions as a narrator, telling the story of the play. In Act 1 Scene 2, Prospero narrates the back story of the play to Miranda. In Act 5, Prospero also serves as a narrator, in a sense telling the story of the play and making...

There are several ways in which the character of Prospero in The Tempest by William Shakespeare functions in a manner analogous to the playwright himself.

First, Prospero at some points functions as a narrator, telling the story of the play. In Act 1 Scene 2, Prospero narrates the back story of the play to Miranda. In Act 5, Prospero also serves as a narrator, in a sense telling the story of the play and making meaning for the characters. Shortly before the epilogue, Prospero says that before the human characters depart for Naples, he will recount:

... the story of my life

And the particular accidents gone by

Since I came to this isle

In other words, he explicitly is portraying himself as a story teller, precisely the role of the playwright. 

The next way in which Prospero acts in a fashion similar to the playwright is by possessing knowledge inaccessible to the other characters of the play. In a sense, an author or playwright is always omniscient with respect to the characters of an imaginative work, and only reveals such knowledge to individual characters or the audience for his own literary purposes. In the same way, Prospero is the only character in the story who has full knowledge of everyone's identity, the history behind the happenings of the play, and what goes on in his island kingdom, only gradually revealing this information to the audience and characters. 

Additionally, both Prospero and the author of a literary work have supernatural powers with respect to the universe within the work. Just as an author can control every aspect of a fictional universe, deciding when it will rain, whether the winds will rise and wreck a ship, or even manipulate time, so too Prospero states in Act V Scene 1:

... I have bedimm'd
The noontide sun, call'd forth the mutinous winds,
And 'twixt the green sea and the azured vault
Set roaring war:

Finally, just as an author controls the characters in a story, Prospero controls spirits through his magic, using these to manipulate the characters of the play, and move the plot to its resolution.