How is Ophelia's madness portrayed?

Ophelia's madness is portrayed through her detachment from immediate reality—those surrounding her in the court—and the dreamlike singing she does. I deliberately include the term "immediate" above because she is, tragically, still aware of the background of what has transpired, and her singing is an alienated way of communicating this to the others.
Even before she begins singing Ophelia makes what I would interpret as a sarcastic jab at Gertrude:

Where is the beauteous majesty of Denmark?

The "sane" Ophelia would never have said this, and the sarcasm shows that Ophelia knows Hamlet's whole situation is at least in part caused by Gertrude's devastating him in marrying Claudius. Ophelia's first song is a reference to her dead father Polonius. But the second song is the most poignant (and best-known to readers and audiences):

Tomorrow is Saint Valentine's day,
All in the morning betime,
And I a maid at your window,
To be your Valentine.

The "prettiness" of this is, of course, ironic. In the second stanza it becomes clear that it is her own story Ophelia is telling:

Then up he rose, and donned his clothes,
And dupped the chamber door.
Let in the maid that out a maid
Never departed more.

Many readers have interpreted that Ophelia had "probably not" slept with Hamlet. But the song, in my view, makes it rather obvious that she is the "maid that out a maid / Never departed more." When she continues her singing we see further evidence of it:

Quoth she, "Before you tumbled me,
You promised me to wed."
He answers, "So would I ha' done, by yonder sun,
An thou hadst not come to my bed."

This at least implies that Hamlet rejected her, did not propose marriage, because he had already slept with her. In the lines she speaks after this, Ophelia seems to regain some degree of lucidity in speaking of her father's death and her awareness that the absent Laertes must be informed of it. But these words are said in a remote manner and are interspersed with detached phrases such as "Come, my coach" and the slightly inappropriate (under the circumstances) "Good night, sweet ladies."
Altogether Ophelia's "insanity" has driven her into a private world, but also made her realistically view things and say things in front of others she would not have revealed before. It is, of course, possible that in her song she's simply imagining she had gone to bed with Hamlet, or expressing what she had only wished to do. But I believe this is a case where in her disordered mental state she is telling the truth and unlocking one of the keys to the whole tragedy of the play.