There is some concern about the ghost as a result of the religious debates that existed during Shakespeare's day. England was largely Protestant, and there are no ghosts in Protestant beliefs; the souls of the dead go to either Heaven or Hell, and they cannot return to earth. Some people did believe that demons could appear in the guise of the dead and, in this way, tempt otherwise good people to do terrible things (like kill their uncles). Catholics, on the other hand, believed in Purgatory; a soul could, theoretically, return from this region of the afterlife. The ghost does say that he is
Doom’d for a certain term to walk the night,
And for the day confined to fast in fires,
Till the foul crimes done in my days of nature
Are burnt and purged away . . .
This does sound an awful lot like Purgatory, where a generally good soul would go on its way to Heaven in order to be purged of the sins of which it had not been absolved when the soul's body died. Wittenberg, however (where Hamlet and Horatio go to school), is very Protestant. It was the site of Martin Luther's rebellion against the Catholic church in 1517. Therefore, this could be why Horatio is suspicious of the ghost.