Hypocrite: He advises the players to resist theatrics and appears to revere passionate people like Fortinbras, but he seems to like grand gestures like the play within a play and is unable to act on his passionate impulses.
Karma Houdini: Sort of. He kills Polonius, and although Claudius tries to have him killed on the quiet Hamlet evades punishment. He also seems to receive no punishment for the deaths of Rosencrantz, Guildenstern OR Ophelia, until the very end, and it's implied that Laertes's forgiveness absolves him completely.
Tragic Mistake: Hamlet's downfall can be traced back to the moment where he sees Claudius at prayer and decides to wait until later to avenge his father.
Writers Cannot Do Math: Leading to problems with calculating his age. While he is a student (which at the time wouldn't have put him past his late teens or early twenties), the gravedigger indirectly mentions that he is about thirty. Real-life casting issues may have played a role here.
Comforting the Widow: Claudius "comforts" Gertrude. It helps win him the throne. On the other hand, he does seem to genuinely love her. It is not an unpopular Alternative Character Interpretation that the throne was an afterthought and Claudius killed the king solely for Gertrude.
Ignored Epiphany: Actually comes close to repenting at one point, but he can't make himself give up the kingdom. Instead he gives up any thoughts of atoning and just concentrates on being King.
Not So Different: His actions and those of Hamlet, specially regarding the letter to England, are rather similar.
Sibling Triangle: Claudius murders his brother and marries his brother's wife. Interpretations vary as to how complicit Gertrude is in the plot.
Disposable Love Interest: She's completely passive and entirely defined by her relationships with the male characters. When they all disappear (Hamlet rejects her, her father dies and Laertes is abroad) she goes completely to pieces and commits suicide.
Does Not Like Shoes: In many adaptations — theatrical productions, films, paintings, etc. — Ophelia is barefoot during the mad scene.
Driven to Suicide: Probably, unless she was just so insane she didn't take heed to her clothes soaking up water while lying in a river. This is all, of course, open for debate.
Speech Impediment: In certain interpretations, Ophelia does have a lisp, and some of her lines actually reflect this (for example, "twice two months" is understood as "two-es...two months). This gives Hamlet's line ("...you lisp, you nickname God's creatures...") a second, literal meaning.
Chekhov's Gunman: He is introduced in the first act and appears briefly in two scenes before departing for most of the play. He returns to play a major role in the final act.
Foil: For Hamlet, who Lampshades it in the final scene. Both seek to avenge their fathers' deaths, but while Hamlet is extremely cautious to establish his uncle's guilt, Laertes almost commits regicide based on nothing but a hunch.
You Killed My Father: His reaction to his father's death stands in stark contrast to Hamlet's behaviour. While Hamlet's hesitation to seek revenge proves to be his undoing, Laertes is quite the opposite. After hearing that Polonius has been killed, he immediately jumps to the conclusion that Claudius is responsible. He returns to Denmark with an army of supporters, bent on committing regicide.
Abusive Parents: Certain interpretations of Polonius show him as this towards Ophelia, manipulating her and keeping her emotionally stunted.
Berserk Button: Fairly meek and submissive for most of the play but insulting or threatening Claudius will make her go Incredible Hulk on your ass. "Oh this is counter you false, Danish dogs!"
Christmas Cake: When an actress is past her prime, she is said to be "playing Gertrude" (ie: rather than Ophelia).
Guess Who I'm Marrying?: Hamlet is deeply shocked that she marries her former brother-in-law so soon after her husband's death.
Heroic Sacrifice: Some interpretations hold that she knowingly drinks the poisoned drink intended to kill her son.
Parent with New Paramour: She marries Claudius after the elder Hamlet's death. Whether or not she was complicit in the murder varies depending on interpretation.
Karma Houdini: Arguably - some readings of the text and some adaptations have him attempting to conquer Denmark underhandedly rather than just passing through with his army as he claims, and the ending for him is Hamlet supporting him to be the next king. If this was his plan, then he's not only not made to pay for his treacherous actions, he ends up being rewarded for it.
Ho Yay or Heterosexual Life-Partners: He and Hamlet seem quite close, to the point of Horatio being willing to commit suicide upon Hamlet's death, and some productions add a homoerotic element to their relationship.
I Should Write a Book About This: Hamlet commands him to keep his memory alive after he is dead, and it is implied that Horatio becomes the narrator of the story.
Morality Pet: Probably the only person Hamlet is consistently nice to.