Hamlet is, indeed, an only child. Proof of this lies in his conversations with Horatio and the gravedigger when he reflects on growing up under the care of Yorick. Never once does he mention having a sibling. In fact, it would be unlikely that his parents would have other children due to the confines of their marriage - King Hamlet was often away at war or with other "kingly" duties. Gertrude was home alone most of the time.
There are no siblings mentioned in the play, so it can be assumed that Hamlet is an only child. Claudius only seems to be concerned about Hamlet's response to the king's death and Claudius' hasty marriage to Hamlet's mother. If there were other children involved, Claudius would be less likely to single one of them out. The fact that Hamlet names Fortinbras as the new king upon his death leads the reader to believe that there are no siblings to pass the crown on to... with Claudius' death, Hamlet becomes king. But with Hamlet suffering a fatal, poisonous wound, he quickly names someone to succeed him before dying.