Richard, the hunchbacked youngest brother of the King is nostalgic for times of war and chaos. He would also like the crown. At his basest, he's a serial killer by proxy. Richard is sixth in line for the succession of the crown, which he uses an excuse to commit five murders.
The first half of Richard III shows the manipulations and machinations Richard enjoys while murdering his way to the throne. Not only does he dispatch with the five others ahead of him, but there are many other associates who don't support him that Richard has murdered as well. Once Richard has the crown, his mental state spirals downward as he becomes paranoid with keeping the crown.
Certain elements of this production place it squarely in the horror realm, rather than the historical drama that most other adaptations do. The opening scene show Richard, his shirt off, his spine, deformed and twisted, as the camera follows the spine up and around his shoulder like a snake, leading to its head, which happens to be Cumberbatch's dignified, yet sinister look, as he delivers the opening monologue. Undoubtedly, this will be the Richard III that sets the standard for our generation.