There is certainly no textual suggestion of a sexual relationship between Hamlet and Horatio. They aren't even together in more than a handful of scenes in the play. Horatio's purpose in the play to serve as an example of a loyal friend, to contrast with the disloyal friends of Rosencrantz and Guildenstern who are really only with Hamlet at the King's command and for the hope of earning a "king's remembrance" (reward) for finding out what is wrong with Hamlet.
Horatio has been at Elsinore since King Hamlet's funeral--coming to comfort a friend at the loss of his father. He learns of the ghost and takes it upon himself to tell his distressing news to Hamlet. He is a sounding board for Hamlet' anger over the King's carousing. Horario is rightly justified in warning Hamlet about the potential dangers of going off to talk with the ghost. Horatio shows his loyalty when he swears to Hamlet to keep secret everything that transpired the night with the ghost.
We don't see Horatio again until the middle of Act 3 when Hamlet asks Horatio to help him observe Claudius for signs of a guilty reaction during the upcoming play. Hamlet is again relying on the loyalty of a friend. He compliments Horatio for his balance of "blood and judgement," meaning that he admires Horatio for his balance of emotion and intelligence.
In Act 4 Horatio is the recipient of a letter from Hamlet telling him that he has escaped the boat to England and returned to Denmark. In Act 5 he meets up with Hamlet in the graveyard and they talk about all that transpired on the boat. He is there to again warn Hamlet about the potential danger of this fencing match that Claudius is proposing.
In the final scene of the play he devastated by the poisoning of Hamlet and says that he would like to commit suicide, claiming to be more "antique Roman than a Dane." He is alluding to the noble suicides of Cassius and Brutus in the story of Julius Caesar. Hamlet tells him his must live on the explain the story of what happened here and why. This is exactly what Horatio does when Fortinbras arrives. Horatio is the best kind of friend Hamlet could have.