What do we learn about Horatio's character in Act I, Scene 1 of Hamlet?
In Act 1 Scene 1 we learn that Horatio does not believe Barnardo and Marcellus, who claim to have seen a ghost walking around at night near the castle. They beg Horatio to stay so he can see for himself.
The ghost does appear, and the mean ask Horatio to speak to him. This is likely because Horatio holds a higher position and the ghost had not spoken to either of the other men in the previous occurence. They figured it might speak to a man with higher title.
We also learn that Horatio is a friend of Hamlet, for he wants to go and tell Hamlet immediately after they see the ghost:
Let us impart what we have seen tonight
Unto young Hamlet, for upon my life,
This spirit, dumb to us, will speak to him.
Do you consent we shall acquaint him with it,
As needful in our loves, fitting our duty? (I.1.168-172)
Horatio is sure that this ghost, whom resembles the late King Hamlet, will surely speak to Hamlet since he will not to any of them. He wants to relay the information out of duty and love for the young prince so we also learn that Horatio has a good relationship with Hamlet.