In Richard III How does William Shakespeare make the audience feel dislike towards Richard, but also sympathise with him?
This is an excellent observation, and it is key to note Shakespeare's accomplishment in creating a truly villainous character that nevertheless manages to engage us and draws us as an audience to him with his charisma and manipulation. In spite of the way that he is always very open about his villainous and evil schemes, his fascination and magnetism draw us to him. In a sense, of course, we fall into the trap that so many other characters fall into by being taken in by Richard's blandishments and presentation. We, like characters such as Lady Anne, even though we know he is evil through and through, are seduced by his presentation and wordplay, and yield to his charms and powerful character.
You might want to examine Richard's monologues to analyse this aspect of hsi character further. The way in which we as an audience are shown Richard's evil nature and his devilish schemes through these monologues should make us hate him, but instead Shakespeare uses them very subtly to manipulate us into admiring him if not liking him, just as Richard is able to do to other characters. Consider Act I scene 1, in which Richard places the blame of his evil nature and the hatred that he bears for others on the way that others have treated him because of his deformity. This speech cunningly presents the other characters of the play as villains and Richard as an innocent victim, helping us to sympathise with Richard's character. Of course, as the play develops, we realise that we are being manipulated.