In Shakespeare's plays how many suicides are recorded?

13 suicides total.

Romeo and Juliet - 2
Julius Caesar - 3
Othello - 1
Hamlet - 1
Macbeth - 1
Antony and Cleopatra - 5

http://absoluteshakespeare.com/trivia/facts/facts.htm

There appears to be some disagreement between sources! Here is a list of 13:
"Suicide occurs an unlucky thirteen times in Shakespeare’s plays. It occurs in Romeo and Juliet where both Romeo and Juliet commit suicide, in Julius Caesar where both Cassius and Brutus die by consensual stabbing, as well as Brutus’ wife Portia, in Othello where Othello stabs himself, in Hamlet where Ophelia is said to have "drowned" in suspicious circumstances, in Macbeth when Lady Macbeth dies, and finally in Antony and Cleopatra where suicide occurs an astounding five times (Mark Antony, Cleopatra, Charmian, Iras and Eros)."

http://absoluteshakespeare.com/trivia/facts/facts.htm

But this list does not include Goneril (King Lear) and Timon (Timon of Athens), who are included in another list (previously available, now offline), which showed these 12:
Goneril (King Lear)
Juliet (Romeo and Juliet)
Lady Macbeth (Macbeth)
Cleopatra (Antony and Cleopatra)
Brutus (Julius Caesar
Mark Antony (Antony and Cleopatra)
Cassius (Julius Caesar)
Ophelia (Hamlet)
Othello (Othello)
Portia (Julius Caesar)
Romeo (Romeo and Juliet)
Timon (Timon of Athens)

Taking them together, there may be up to 15! Or, counting only those that are agreed on by both list-compilers, there may be as few as 10.

And don't forget the play of Pyramus and Thisbe within a Midsummer Night's Dream, in which both title characters kill themselves, only to have the actors who play them immediately stand right back up...