How is Richard III's character developed throughout the play, referring to specific incidents and language?

Review the teacher's prompt - does your teacher want you to explain how Gloucester / Richard III develops as a character, a leader, or to explain how his "character", his personhood, develops? This might help to focus your answer.

First, read about the War of the Roses and the houses of York and Lancaster to get a feel forthe contextof the play.  You will thus begin to understand the importance of succession, precedence, and order that are so valuable to Shakespeare's period.  Roman numerals are used to designate European monarchs because many monarchies and aristocratic lineages were established in a period when Latin was deemed the language of power and influence.  The use of ordinal numbers indicates a description of the monarch in order of succession, in this case, Richard THE THIRD. 

Shakespeare valued order; his use of its opposites, treason, usurpation, homicide and outlawry indicate this. But there are other aspects of Renaissance life that give Gloucester / Richard III his "character."  For one thing, during this period of religious and political upheaval, mainly between Protestants and Catholics, one could not always look to succession, order and precedence as coming from God but instead see legitimacy as arising from the natural gifts that humans themselves could cultivate - namely, reason, logic, strength and personal courage, and mental fortitude.  The value of these traits in the Renaissance can be seen in the advancement of politics, trade, warfare, sea faring, and the arts and sciences.  

There is a dark side to this and Niccolo Machiavelli's book "The Prince" explains how the good Prince (and by good he means the Prince that can extend his rule) must rely on devious means to retain power thus expressing the Renaissance value called "virtu."  Virtu is a play on the idea of Virtue, a moral quality, but instead of being good and honest and humane, the new Prince of "virtu" must be able to do the opposite.  

This is all very relevant to Gloucester / Richard III as a character.  For instance, what does he do to assert himself to gain control of his own fortune rather than leaving it all to chance? Where does he exercise strength, intelligence, prowess and courage, all requisites of a Prince? What above all other things does he want for his people?   But also remember that a successful Prince can't be too hated because that will lead to civil strife. 

The concepts of self - knowledge and self - mastery might form the basis of an examination of the development of Gloucester / Richard III's "character" as a person.  Sophrosyne, balance and their opposite, hubris were employed by Shakespeare. Does Gloucester / Richard III learn about himself in this play?  If you notice a shift in your sense of how he "develops", then you have a paper. 

Start with your three favorite passages from the play - say his wooing of the wife of the man he just had murdered in the first scene - then go to the end of the play and examine his speeches before he dies and his dreams and state of his conscience.  A good rule of thumb is to come up with three very good examples of what you are trying to proove.