How do loyalty, betrayal, and depression affect Hamlet?
All of these are major themes in Hamlet and have very strong effects on Hamlet.
Loyalty: Loyalty to his father drives him to seek revenge against his uncle, Claudius, who murdered King Hamlet. What is sad, though, is that Hamlet has so few people who are loyal to him. Horatio is really the only one that Hamlet knows he can trust to be loyal to him.
Betrayal: Betrayal has an enormous affect on Hamlet. He feels betrayed by his mother, Gertrude, for marrying so quickly after her husband's death and for marrying her own brother-in-law. He feels betrayed by Ophelia, who is a puppet in the hands of her father and brother, Polonius and Laertes. Hamlet endures grievous betrayal by his university friends Rosencrantz and Guildenstern who are led by Claudius into spying on him.
Depression: Depression is something that affects, even plagues, Hamlet from the very beginning of the play. He is depressed at his father's death, depressed at his mother's hasty remarriage, then more depressed (and shocked and enraged) at the news that his father was murdered. Hamlet struggles against depression even after he is set upon his determined course to take revenge against Claudius.
Other characters, too are affected by the calamities that depress Hamlet, which come about through betrayed loyalty. Ophelia comes to mind and, though to a lesser extent, Horatio.
Both of these characters are very close to Hamlet. Being close to someone who suffers, it is inevitable that some of affects will spread over to the other's life.
Ophelia is driven mad by Hamlet's sufferings. She feels rejected and betrayed by Hamlet, then, as she is forbidden to see him by both her father and brother, she begins to feel depression. When Polonius is murdered by Hamlet, she sinks--literally and figuratively--into the deep dark depression from which there is no return.
Horatio feels Hamlet's betrayal by Gertrude and Claudius. He is so taken by the death of Hamlet that he swears to drink the poison and die alongside Hamlet, though Hamlet stops him by requesting Horatio speak for Hamlet. Horatio, the only true friend of Hamlet, does not drink the poison once the Prince of Denmark has asked him to remain alive in order to pass the kingdom to Fortinbras and to tell Hamlet's true story.