What can be gleaned from Shakespeare's drama about one of its protagonists, Cleopatra?

Cleopatra was a strong woman who was in an extremely difficult position and tried to use the tools available to her to maintain her precarious grasp on power as long as she could. 
The thing you have to remember is that while Cleopatra was extremely important to the people of Egypt as the Pharaoh sitting in Alexandria, she was in a precarious position constantly.  Her position of power was limited or threatened from within and without.  Julius Caesar had put Cleopatra on the throne, and then she had a son with him, Caesarean, that allowed her to keep it.  
As far as the Romans were concerned, Egypt belonged to them.  When the triumvirate divided up the Roman holdings, Antony ended up with Egypt.  He happened to be quite the womanizer, so it was a bad combination.  Cleopatra might have used Caesar to come to power, and but used Antony to try to stay in it. 
At this time, the Roman world was divided between Antony, Octavius Caesar, and Lepidus.  Shakespeare describes, through Enobarbus, how Cleopatra seduced Antony.  Enobarbus describes Cleopatra’s elaborate barge to Agrippa and Macaeanas, Octavius Caesar’s henchmen. 

DOMITIUS ENOBARBUS
I will tell you.The barge she sat in, like a burnish'd throne,Burn'd on the water: the poop was beaten gold;Purple the sails, and so perfumed thatThe winds were love-sick with them; the oars were silver,Which to the tune of flutes kept stroke, and madeThe water which they beat to follow faster,As amorous of their strokes. (Act 2, Scene 2)

Cleopatra’s vivid personality extended beyond the fancy barge she used to hook Antony.  Her scenes with Antony often involve arguments, and if she is not arguing with him she is worrying about him.  Cleopatra’s messengers are a perfect example.  She sends them to find out about Octavia, Antony’s new wife, and has an emotional reaction to their answers. 
Cleopatra needs Antony, but she realizes that she is losing him.  Octavius Caesar tried to gain control of him by marrying him to Octavia, his sister.  Cleopatra was aware of that.  After Antony cuts ties with Octavius, it gets even worse.  His military might is slipping and Cleopatra knows it.  She feels that Antony is losing his perspective.
Cleopatra knows that the end is coming.  She realizes that she will likely have to kill herself, and she makes preparations.  She tries to protect her children from Octavius, especially Caesarion.  These actions demonstrate Cleopatra's strength and presence of mind.  She even encourages Antony to commit suicide, which the Romans consider a noble act, by tricking him into thinking she is dead.
Her own suicide results in Shakespeare's version from a conversation with Octavius.  You could also say that Cleopatra waited to see what she could accomplish with Octavius.  She could not seduce him the way she did Julius Caesar.

CLEOPATRA
My master, and my lord!OCTAVIUS CAESAR
Not so. Adieu.Flourish. Exeunt OCTAVIUS CAESAR and his train
CLEOPATRA
He words me, girls, he words me, that I should notBe noble to myself: but, hark thee, Charmian. (Act 5, Scene 2)
In the end, Cleopatra refused to be paraded through Rome as a trophy of Octavius Caesar, but that is what would happen to her children.  She was a strong, confident woman.  She ended life on her own terms.