A different interpretation would be as follows. Hamlet tells Horatio:

Report me and my cause aright
To the unsatisfied.

HORATIO
Never believe it.
I am more an antique Roman than a Dane.
Here's yet some liquor left.

Horatio intends to commit suicide in the manner of ancient Romans and follow his friend. He picks up the cup containing what is left of the poison drunk by Gertrude and forced onto Claudius, but Hamlet wrests it away. When he asks Horatio to

Absent thee from felicity awhile

he means that death is felicity and is asking his friend to remain "in this harsh world / To tell my story." Hamlet pleads:

O God, Horatio, what a wounded name,
Things standing thus unknown, shall live behind me!

Horatio follows Hamlet's instructions and is able to offer a summation to Fortinbras and to the members of the Danish court which will exonerate Hamlet of any treachery or other wrongdoing and serve as a sort of epilogue to this long and complex play.

In these lines Hamlet is dying and speaking to his friend Horatio. Hamlet is asking Horatio for a favor. In the first line, Hamlet envokes their friendship. He asks Horatio to remove himself from his regular pleasures in life for a while. It's like he is asking Horatio to grieve, but more than that the next two lines ask for Horatio to work to spread the news of what has happened to Hamlet and his family. Essentially, Hamlet is asking Horatio to sacrifice his own time to report to the world the tragedy that happened in Denmark so that it doesn't happen anywhere else again.