How did Shakespeare change the historical record in writing Richard III to suit his needs dramatically? as in, what liberties did he take in order to serve his "agenda" in presenting the facts in a way that the audience could understand?
I wrote a response to a similar question here, if you’re interested in Shakespeare’s dramatic transformation of King Richard.
If you want to know about the historical Richard, then this is a better question for a historian than a literary critic. I don’t quite know what you mean about ‘agenda’ other than the agenda of dramatic effectiveness, and the audience would have been capable enough of understanding whatever they saw if the other plays are any guide… but I can answer you as far as the historical sources that were available to Shakespeare is concerned.
The play is mostly very faithful to its sources, Sir Thomas More’s The History of Richard III, which only deals with Richard’s rise to the throne; Edward Hall’s The Union of the Two Noble and Illustre Families of Lancaster and York, which includes an adaptation of More’s account; and Holished’s Chronicles, which adapted Hall’s version, and Hall’s adaptation of More. In these texts, Richard is described as deformed, as having taken power from his nephews, who he imprisoned and then probably killed. There is a description of the way he acts in front of the populace in order to secure their support, his coronation, the death of Hastings, Buckingham’s rebellion, and the details of how he came to be overthrown by Richmond. More is less certain about how much Richard can be blamed for Clarence’s death, but the essential information is all there. Whether or not the original ideas were derived from old propaganda, Richard as a tyrant and child-murderer was accepted as fact in Shakespeare’s time.
So the main additions are the dramatic fleshings-out of the narrative, and the specific words and thoughts that Shakespeare gives his characters, for instance how Richard employs the assassins, and how he woos Lady Anne, and how all the characters feel about these events.
One big divergence is the existence of Queen Margaret. Historically she had died by the time that Richard came to the throne, but partly because Richard III is a sequel to the Henry VI plays, and partly because she has a dramatically significant role as a prophetess, Shakespeare retains her as a chorus character that keeps the continuity going between the four plays.
The smaller things include the depiction of particular minor characters. Lord Stanley was an ancestor of the theatre company’s patron, Lord Strange, and the character is portrayed in a favourable light, being unsure of Richard’s rightness and so on, where, for instance, the historical Lord Stanley crushed Buckingham’s rebellion for Richard and received Buckingham’s land from Richard in return. Probably for a similar reason, Pembroke and Herbert, the ancestor’s of Lord Pembroke, who had a theatre company Shakespeare’s play might have been acted by, are amongst the Lords associated with Richmond’s cause that Shakespeare singles out (they were there, but Shakespeare emphasises them).
If you’re very interested in the main source, the Richard III society has the whole of Holinshed’s account of Richard III on their website for free.