I would content that Cleopatra's death is not the climax of the tragedy. Rather, the climax occurs when Antony's naval forces surrender to Caesar in Act 3.10.
Furious at the loss and convinced that it occurred because of Cleopatra's betrayal, Antony begins his journey home. By Act 4.13, however, Cleopatra, has decided (on the advice of Charmain) to lock herself away in her monument. She instructs Mondain to get a message to Antony, telling him that she has committed suicide. Additionally, Cleopatra emphasizes to Mondain that he is to make sure that Antony feels guilty for her decision:
To the monument!--
Mardian, go tell him I have slain myself;
Say that the last I spoke was 'Antony',
And word it, pr'ythee, piteously: hence, Mardian;
And bring me how he takes my death.--
To the monument!
Obviously, Cleopatra has no real intention of carrying through with her threat. Like most of her actions, her intent is to direct Antony's attention to where she feels it rightfully belongs: on herself.
Word reaches Antony, who commits suicide. When Cleopatra learns of his death, she clasps an asp hidden in a fruit basket, to her chest and is killed by the snake's venom (although one suspects that her decision is motivated more by the promise to be paraded through the streets by Caesar's forces than by grief over Antony.)