Three things that are described as unnatural in act 1 of Hamlet are the atmosphere, the relationship between Claudius and Gertrude, and the death of Hamlet’s father.

After sighting the Ghost in scene 1, Horatio declares,

In the gross and scope of mine opinion
This bodes some strange eruption to our state.

This sense of foreboding emphasizes and portends impending disorder in the “state” or royal family of Denmark. Horatio compares the appearance of the Ghost to the time before Julius Caesar’s death when

The graves stood tenantless, and the sheeted dead
Did squeak and gibber in the Roman streets;
As stars with trains of fire and dews of blood,
Disasters in the sun; and the moist star,
Upon whose influence Neptune’s empire stands,
Was sick almost to doomsday with eclipse.
And even the like precurse of feared events,
As harbingers preceding still the fates
And prologue to the omen coming on.

These unnatural events in nature—empty graves, talking and walking corpses, blood in dew, and a nearly eclipsed (“sick almost to doomsday”) moon—presage the disaster and tragedy which unfold in the play.

In scene 2, Hamlet describes the relationships between himself and Claudius as well as Claudius and Gertrude as unnatural. After Claudius addresses Hamlet as his cousin and son, Hamlet mutters,

A little more than kin and less than kind.

Claudius, brother of old King Hamlet (Hamlet’s father), quickly married Queen Gertrude (Hamlet’s mother) after King Hamlet’s death. Now Hamlet is both Claudius’s nephew and son (or stepson)—making their relationship not only kin but also “less than kind” or less than natural. Also, Claudius is not only Gertrude’s former brother-in-law, he is now her husband! Their relationship is certainly “more than kin”—connected in too many ways and too closely—and unnatural.

In scene 5, King Hamlet's Ghost tells his son to avenge his death, which was a

murder most foul, as in the best it is.
But this most foul, strange, and unnatural.

His death was unnatural; Claudius poured poison into sleeping King Hamlet’s ear. This murder was “foul” in so many ways—Claudius stole the king’s throne as well as his wife. Now this usurping murderer rules the kingdom and country of Denmark. Therefore, King Hamlet implores Hamlet to avenge his murder, saying,

Let not the royal bed of Denmark be
A couch for luxury and damnèd incest.