In Hamlet what are some lies that have been told?
The biggest lie that was told was that Claudius was the king, only after murdering his brother. He obviously had to lie in order to usurp the position. He lied to the kingdom, to Hamlet, and to Gertrude--we don't get his exact denial of any wrongdoing, but, in the opening scenes he acts all grieved, and mourns for his brother's death. He even tries to help Hamlet through his grief. His entire life is a lie, one that began as soon as he started scheming to become king.
Another lie that is told is when Hamlet confronts Ophelia, and tells her bluntly and harshly, "I loved you not." This lie is later refuted as Hamlet comes home to find Opehlia dead. At this point, the truth comes out as he jumps in her grave and cries out in grief,
"I lov'd Ophelia; forty thousand brothers could not, with all their quantity of love, make up my sum."
So, for whatever reason, he lied to Ophelia's face about his love for her.
Rosencrantz and Guildenstern originally lie to Hamlet about their purposes in coming to visit--they say they are here for the wedding, funeral, and to be his friends. However, the truth is that the king asked them there to "spy" on Hamlet, and to keep an eye on his strange behavior. At the end of the play, Laertes and Claudius are anything but forthcoming about their devious "poison" plot.
Lies abound in Hamlet, and come from all directions, and from most of the people involved. I listed just a few here, but I hope that can get you started; good luck!