There are several warnings in the play. Young Fortinbras warns Claudius that he plans to attack Denmark with his newly built army. The ghost warns Hamlet of his murder, and spurs him to revenge. Horatio warns Hamlet not to follow the ghost, fearing it to be a demon of some kind. Polonius and Laertes warn Ophelia not to associate with Hamlet anymore, fearing that he will hurt her or leave her. These are just a few examples in a play filled with backstabbing, sabotage, surveillance, plotting, and intrigue.
As far as madness, Hamlet stands as an example. Although his is certainly fake, at least to begin with. He converses with both Ophelia and Polonius in a manner that suggests psychological issues, but nearly all of his lines have double meanings. For example, when he speaks to Polonius, he mentions "Jephtha", a biblical allusion to a man who inadvertently sacrificed his daughter. A perhaps not so subtle hint....Ophelia, on the other hand, definitely goes mad. It's a combination of Hamlet's treatment, her father's oppressive nature, and her brother's attempt to control her. Oh, and the fact that her lover killed her father. Essentially, she shows the condition of women in a society where they have no power, and suffer at the mercy of the men in their lives.
Finally, the easiest one: death. Well, pretty much everyone except Horatio and Fortinbras die at some point in the play. Some die early on (Polonius, Ophelia); others die at the end (Hamlet, Gertrude, Laertes, Claudius). Some die offstage, outside the direct action of the narrative (King Hamlet, Rosencrantz and Guildenstern, Yorick). These deaths still play an important role in the play however. There's even metaphorical death, in the state of Denmark itself. The play is rife with references to disease, illness, and dying, all connected to the nation and its people.