What role does dishonesty play in Hamlet?I'm also wondering how this changes his actions in finding loyalty in others.

Dishonesty definitely plays a major role in both Hamlet, the protagonist and Hamlet, the play. Though the tragedy ensues from Hamlet's prolonged indecisiveness, his vacillating mood, yet one simply cannot be evasive of the dishonesty of Claudius that propels hamlet towards his tragic fall.

The murder of his father in the hands of his uncle, Claudius, renders existence difficult for Hamlet. To add to his wounds, Gertrude re-marries hastily giving vent to their lustful unjust characters. This dishonest, heinous deed on the part of Claudius once revealed to Hamlet by his father's ghost creates a tumult of emotions. Yoked with this is his mother's lack of sorrow felt for the dead husband. Such overpowering emotions create a furore within Hamlet and he commits the twin crimes of murdering Polonius and being the reason behind Ophelia's madness and her final death. His love for Ophelia is walloped by his unusual love for Gertrude as well as the pall of injustice that envelopes Denmark.

It is dishonesty that prevails from the beginning to the end of the play. Most of the deaths take place due to the lack of loyalty. The King is murdered by his disloyal brother, the disloyal wife marries the disloyal brother. Laertes makes use of dishonest means to win the duel, which at the end leads to the fatal ends of Hamlet, Laertes as also Gertrude, who drinks the poisoned wine intended for hamlet, again by Claudius.