How does the use of rhetoric in Shakespeare's Richard the Third demonstrate the themes of the play?
Well, first of all, there are many themes in Shakespeare's King Richard the Third: war, justice, manipulation, betrayal, fate, appearance vs. reality, etc. However, since you asked specifically about rhetoric, it is MOST appropriate to discuss the theme of manipulation as that is the quality most portrayed by the main character Richard.
First, how is Richard manipulative? Well, his main goal is to gain power, and because Richard is Edward IV's very youngest sibling, Richard is not likely to become king in a "normal" way. It matters not what way Richard obtains that power. Obtaining power in a moral and honorable way isn't important. Richard discovers pretty quickly that he needs people on his side in order to achieve his ends and in one of the most eloquent pieces of rhetoric in Shakespeare, Richard decides to use the Bible in order to manipulate!
But then I sigh; and, with a piece of scripture,
Tell them that God bids us do good for evil:
And thus I clothe my naked villainy
With old odd ends stolen out of holy writ;
And seem a saint, when most I play the devil.
Manipulation is clear here, as we hear Richard's thoughts. He is purposely using the bible in order to "tell them that God bids us do good for evil." He is pretending to be holy while he is truly devilish in his manipulation of the people. These are Richard's plans, his thoughts, and not his explanation to the people. He needs to be the master of secrecy so that the people don't discover his deviousness. There is no better example of rhetoric used for manipulation!
Manipulation is also shown through Richard's actions (in addition to rhetoric). In fact, these actions are so deplorable that Richard has to return to rhetoric in order to convince the people otherwise. Richard's older brother (George) is most certainly in the way of the throne, so Richard gets him incarcerated (unjustly). Richard then engages in a marriage for political reasons (to Anne). The next logical, manipulative step is to have Clarence killed, so Richard hires men to do that (followed by hiring the same men to kill others in the way). It is when Richard begins to kill children that the people begin to get a clue that Richard isn't all that he seems. Richard then needs to prove he is NOT what people suspect.
I can add colors to the chameleon
Change shapes with Proteus for advantages
And set the murderous Machevil to school.
What a beautiful piece of Shakespearean rhetoric! "I can add colors to the chameleon." Richard becomes a shapeshifter.
The irony is, in all of his planning, scheming and manipulation, he ultimately fails. HOWEVER, Richard's final words are still a beautiful show of Shakespearean rhetoric when he is knocked off his horse as he is deserted by his own people in the Battle of Botsworth Field.
A horse, a horse, my kingdom for a horse!
Ah, the climax of the play! Richard is killed. Goodbye Richard... all of your amazing rhetoric couldn't save you!