The novel The Picture of Dorian Gray can be read many ways, but I think it begs to be read as metafiction (writing about writing).  More accurately, it is criticism about criticism.  We have an artist, Basil; a critic, Lord Henry; and the art, Dorian.  All three match up with the artist, Wilde; the critics, (Robert Boyle is a good one); and the art, The Picture of Dorian Gray.  Are there any connections to be made among these sets of threes?  What are Wilde's aims in art and for the artist?

Three is indeed a magic number in the novel and in art.  There are three levels of "beauty" as first defined by many Romantic critics: the picturesque (least), the beautiful, and the sublime (most).  Maybe you can examine these levels of beauty in art, nature, and human nature in the novel.

Wilde separates his thoughts on art in two opposing personas: the Romantic Basil and the Realist Lord Henry.  Caught in the middle is the art, Dorian, who becomes a kind of Faustus archetype who makes a pact with the devil.  Is the goal of art to live forever?  Is this due to the vanity of the artist?  You may do a Jungian, Freudian, Mythological, Archetypal reading of the novel based on these concerns.

You should read the excellent criticism on the Dorian Gray Enotes page by Susan Boyle (see below; you may need a subscription to access it fully).  She analyzes James Joyce's critical analysis of the novel in which he says:

Wilde seems to have some good intentions in writing it—some wish to put himself before the world—but the book is rather crowded with lies and epigrams. If he had had the courage to develop the allusions in the book it might have been better.

Hope this helps.  Good luck!