It’s a little tricky when you’re talking about the mouth and voice of a dead man. It evokes a kind of morbid ventriloquism. It is hard not to picture a dead Hamlet’s mouth being animated and voiced by Horatio. Horatio will prop him up on his knee, move his lips for him and say, “He has my dying voice!”
Of course that’s not what is actually going on here. It’s just Horatio saying he’s going to be able to able to drum up a bunch more support for Fortinbras’ claim to the throne when he reports what Hamlet had to say. It’s just – how he says it.