"Horatio, I am dead, thou liv'st; report my cause aright to the unsatisfied". Contextual relevance of Hamlet's request? Contemporary relevance?...in Shakespeare's Hamlet.
In Shakespeare's Hamlet, Act Five, scene two, Hamlet has been poisoned by Laertes in Claudius' plot to have the Prince of Denmark killed.
Things quickly spiral out of control once Gertrude drinks the poison intended for Hamlet. As she dies and tells her son what has happened, Hamlet demands that the doors be locked. He has already been cut by the poisoned sword, but he does not know it. Laertes is also dying and clarifies what has happened:
It is here, Hamlet. Hamlet, thou art slain;
No medicine in the world can do thee good.
In thee there is not half an hour of life.
The treacherous instrument is in thy hand,
Unbated and envenom'd. The foul practice(325)
Hath turn'd itself on me. Lo, here I lie,
Never to rise again. Thy mother's poison'd.
I can no more. The King, the King's to blame.
Hamlet learns he is poisoned and there is no antidote. Laertes admits he is responsible and the same sword has poisoned him. He confirms the cause of Gertrude's death, but points his finger of accusation to the King who has orchestrated all of this.
Hamlet runs at Claudius and not only stabs him with the poison-tipped sword, but forces some of the poisoned wine down his throat as well.
The point envenom'd too! Then, venom, to thy work. (329)
Here, thou incestuous, murderous, damned Dane,
Drink off this potion! Is thy union here?
Follow my mother. (332-334)
As Hamlet is dying, he asks Horatio to share news of what has happened:
Horatio, I am dead;
Thou livest; report me and my cause aright
To the unsatisfied. (346-348)
In terms of the contextual relevance, those present in the chamber will need to hear the truth; Horatio is the only one who knows. Everyone else is dead. Recall that when Hamlet stabs his murderous uncle, the assembly cries:
Treason! treason! (330)
Without knowing the details, Claudius (as King) has been stabbed and asked for help. To all appearances, Hamlet is a murderer. These questions will need to be answered for two reasons. First, to put the minds of the court at ease, understanding that Hamlet's father was murdered by his brother. Hamlet's name will need to be cleared of all blame. Second, as Hamlet does not leave an heir, he believes that the throne and Denmark will go to Young Fortinbras. Hamlet's story will clear the way for this to happen. And we already know that when Hamlet's path crossed that of Fortinbras fighting Poland for a patch of land, he found a great deal in the other man to admire.
If I understand the second portion of your question correctly, I would assume that Horatio's "deposition" to the "authorities" would be the only source of proof with regard to what led to Hamlet's murder of Claudius, or explained the deaths of so many others in the Danish court—to the point that no one is left to rule Denmark, and it passes into Norway's hands. Though it would be considered circumstantial to an extent, enough people would have heard Laertes' words to realize that Claudius plotted Hamlet's death and not the other way around.