I agree with Laraine about the "antique Roman" interpretation - he was contemplating suicide. His next line is even "Here's yet some liquor left." referring to Claudius' poisoned cup.

No, Horatio doesn't have much of a personality. He is mostly a vehicle, a way for Hamlet to disseminate information to the audience, a way for other information to be given to the audience through Horatio.

But there are hints of a deep friendship between Hamlet and Horatio; Horatio is the only one Hamlet completely trusts, and as you said, he is friend enough to challenge Hamlet on occasion, and they do exchange words of friendship and devotion.