Read the full text of Richard III Act 3 Scene 5 with a side-by-side translation HERE.
At the Tower of London, Richard makes Buckingham promise that he'll do whatever Richard says.
Catesby enters with the mayor of London. Richard and Buckingham make a great show of declaring their vigilance, saying they're trying to protect themselves (and the mayor) from some impending great danger.
Lovel and Ratcliffe enter with Hastings' head.
When Richard and Buckingham see Hastings's head, they pretend to be sad that Hastings turned out to be a traitor who plotted against their lives.
The lord mayor is convinced that Hastings deserved to be executed without a trial.
Richard says he wishes the mayor were there to hear Hastings's confession so he could tell the commoners what happened. Richard's afraid the citizens of London will be unhappy and blame him.
The mayor runs off to assure the citizens that Hastings's death was actually a rightful and just execution by the state.
After the mayor exits on his errand, Richard reveals the plan he's hatched to gain the crown: he wants Buckingham to go to Guildhall (the seat of municipal government in London) and spread a rumor that King Edward's sons are "bastard[s]" and therefore not the rightful heirs to the crown.
Buckingham should tell people that King Edward slept around a lot and that he once executed a citizen for no good reason.
Buckingham should also say that Edward's mom (who is also Richard's mom) had affairs while her husband was off fighting the French and that Edward is "illegitimate." This would make his kids illegitimate, too, and unfit to be crowned kings.
Richard decides that Buckingham should only hint that Edward was illegitimate. After all, his mom is still alive, and you don't want to shout too loud about your mom's infidelity.
Richard says he's going to hide Clarence's kids and make sure nobody has access to the princes (Edward's children).