One important example of the theme of decay is found in Hamlet's first soliloquy when he says that Denmark, and more specifically his mother's remarriage "Tis an unweeded garden / That grows to seed; things rank and gross in nature possess it merely." Hamlet is comparing his mother and Claudius to unweeded garden. In order for a garden to thrive, be beautiful or bear food, it must be kept free of weeds which will rob the good plants of much needed nutrients. Gardening and weeding is not always easy, but it is necessary. Gertrude may not like being alone or losing her status as Queen, but she should have been stronger and stayed away from Claudius rather than let him and her moral weakness corupt her.
Another example of decay is Marcellus's remark at the end of scene 4. All of the men are out with Hamlet as he prepares to confront the ghost of King Hamlet. The arrival of a ghost is never a good thing -- it is usually to convey unfinished business or to reveal big secrets or, worse case senario, the ghost is really the devil is disguise. In response to this whole situation Marcellus says, "Something is rotten in the State of Denmark." He is fully aware that the ghost is a portent to bad things and this is a potentially very bad situation for Hamlet and for all of Denmark.