Lady Montague

Alas, my liege, my wife is dead tonight.
Grief of my son's exile hath stopped her breath.
What further woe conspires against mine age?

Lord Montague at the Capulet tomb, Romeo and Juliet, Act 5: Scene 3

The Defector
Data played King Henry V in Henry V, Act 4: Scene 1.

Hamlet, MacBeth, Julius Caesar, Richard III

38 plays at least -- A recently confirmed piece The Two Noble Kinsmen isn't included in most anthologies. Love's Labour's Won and Cardenio are amongst the lost works.
154 sonnets as well as two long poems, Venis and Adonis and The Rape of Lucrece
Not all experts agree on which plays belong in what categories. Several defy categorization.
10 tragedies
18 comedies
10 histories

The "problem" plays that might fall into more than one category:

• Cymbeline (comedy or tragedy?)
• Coriolanus (comedy or tragedy?)
• Pericles (comedy or tragedy?)
• Winter's Tale (comedy or tragedy?)
• Troilus and Cressida (history, comedy or tragedy?)
• Anthony and Cleopatra (history or tragedy?)
• Timon of Athens (history or tragedy?)

Of the many books I consulted on this, most neatly skirted the issue by not dividing the plays into categories, prefering to list them in date order or alphabetical order or whatever suited the publishers. I've made the best guess I can with the information available to me.

3 plays: Henry IV, Parts 1&2, and The Merry Wives of Windsor

3 tragedies, 2 histories, and 1 comedy

Well, of course, it depends on your definition of which plays are tragedies. Taking a liberal inclusion of the "problem" plays, the 11 kings are:
Priam -- Troilus and Cresida
Duncan, MacBeth, and Malcolm -- MacBeth
Hamlet, Sr., and Claudius -- Hamlet
Lear and the King of France -- King Lear
Cymbeline -- Cymbeline
Antiochus and Simonides -- Pericles

The Conscience of the King The actors were performing Hamlet.

The Winter's Tale
Pericles
Much Ado About Nothing
A Midsummer's Night Dream
Cymbeline
Coriolanus

Okay, so it's a trick question. Those lines are popular misquotes, so in fact, none appear in the plays. Here are the correct lines.

"The lady protests too much, methinks." Queen Gertrude in Hamlet, Act 3: Scene 2

"Alas, poor Yorick. I knew him, Horatio." Hamlet in Hamlet, Act 5: Scene 1

"But for my own part, it was Greek to me." Casca in Julius Caesar, Act 1: Scene 2

King John offed his nephew Arthur in King John and King Richard had his two nephews killed in Richard III. In the tragedies Claudius was no shining example of avuncular love, either.

Julius Caesar and Othello

Time's Arrow, Part 2
A Midsummer's Night Dream, Act 2: Scene 1

Mercutio and Paris

The line is from Julius Caesar, Act 3: Scene 1. It was used in Star Trek 6: The Undiscovered Country (1991), also the only film to have a line of Hamlet's soliloquy spoken in the "original Klingon."

Tiny Hint: There are 6 plays quoted

Of course, I was hoping you could tell me. Yes, I have identified the lines from Sonnet 147 and Sonnet 18. Perhaps if time permits I will transcribed the entire speech and attempt to pick it apart line by line.

Titus Andronicus was Shakespeare's first play; Henry VIII was his last.

One set in Twelfth Night and two sets in A Comedy of Errors

Midsummer's Night Dream. (1 line)
Henry V (1 line)
MacBeth (3 lines)
Hamlet (4 lines)
Henry IV (2 lines)
MacBeth (2 lines)
Hamlet/Julius Caesar (half a line each)
Hamlet (5 lines)
Henry V (5 lines)
Julius Caesar (1 line)
Henry IV (5 lines)