She married. O, most wicked speed, to post
With such dexterity to incestuous sheets!
It is not, nor it cannot come to good.
But break, my heart, for I must hold my tongue.
Hamlet not only takes issue with his mother's quick remarriage after his father's death, he's also disgusted by the fact that Gertrude is guilty of "incest." (Some critics also speculate that Hamlet secretly wants to sleep with his mother, which you can read about in our "Character Analysis" of Hamlet.)
But first, it's time for a history snack. In Shakespeare's time, incest included marrying your in-laws, not just your blood relatives. So, Claudius' marriage to Gertrude is a pretty big deal —they've broken the church's laws of affinity.
And there's something more particular about the whole marrying-your-brother thing. Elizabeth I, the Queen of England at the time Hamlet was written, was the daughter of King Henry VIII and Anne of Boleyn (Henry's second wife). Henry divorced his first wife, Catherine of Aragon, on the grounds that she had originally been married to his (dead) older brother, Arthur. Henry asked the Catholic Church to grant a divorce on grounds that his marriage to Catherine was incestuous. By making such a big deal out of Gertrude's remarriage, Shakespeare might be doing his part to assure Queen Elizabeth that her mom's marriage was legitimate.