Translate Hamlet's sentiment in the following line, "The time is out of joint: O cursed spite. That ever I was born to set it right."
Hamlet speaks these lines at the conclusion of Act 1, Scene 5. To understand exactly what he means, one has to take note of what has gone on before.
Prior to Hamlet saying this, he had been visited by his father's ghost, which had been haunting the area just below the battlements for some time. Horatio informed hamlet of the visitations. Hamlet then visited the battlements where he eventually encountered the spirit. After identifying itself, the ghost commanded:
Revenge his foul and most unnatural murder.
Hamlet is shocked at this revelation and promises to immediately rush to his revenge:
Haste me to know't, that I, with wings as swift
As meditation or the thoughts of love,
May sweep to my revenge.
Hamlet is devastated when he hears that it was his uncle, Claudius, now his new father and king of Denmark, who had so maliciously killed his father by pouring poison into his ear whilst he was sleeping in the orchard. What makes the crime even more malevolent is the fact that Claudius had seduced the virtuous Gertrude, Hamlet's mother, and married her after he had killed his own brother, a most unnatural act. Furthermore, king Hamlet was not even granted the opportunity to seek divine redemption, for he was murdered in his sleep. Because of this, he was damned to everlasting perdition. All of these should urge Hamlet to a speedy vengeance, which he has promised. As dawn approaches, the ghost disappears.
Hamlet later speaks to Horatio and Marcellus and swears them to secrecy about what they had witnessed. It is then that he speaks the lines quoted in the question. By saying that 'The time is out of joint', Hamlet means that everything has turned topsy-turvy, the natural order of things has been turned upside down, nothing is normal or as it should be. These references stem from what he has just learnt from the ghost and also the current situation in Denmark, which is preparing itself for an invasion by a rebel Norwegian force led by Prince Fortinbras.
In saying 'O cursed spite', Hamlet is suggesting that the forces of destiny have been spiteful and have cursed him. Why? 'That ever I was born to set it right.' He feels cursed about his birth, not about the fact that he has to correct the wrongs in Denmark. The unnatural order has to be restored. Hamlet does not question this fact, but he does question why he was born to do it.
Hamlet has promised to adopt an antic disposition - feign madness - to enact his revenge, but we learn later that he procrastinates to such an extent that many others die as well. This is probably because he was not really that willing to do his duty, in spite of the solemn pledge he had made to his father's ghost.