Fortinbras is important in Shakespeare's Hamlet as a foil to Hamlet, but the play doesn't reveal a great deal of characterization about him.
Early in the play, Claudius reveals in a speech (Act 1.2.17-39) that Fortinbras, having recently lost his father in battle as well as Norwegian lands, to King Hamlet, is pestering Denmark about getting the lands back. Claudius takes diplomatic steps to stop Fortinbras. The prince of Norway here is revealed to be aggressive and perhaps impetuous, in contrast to Hamlet's reasoned, and some might say tentative, approach to a similar situation.
Later in the play, in Act 4.4, Hamlet feels rebuked by Fortinbras, when he sees the Norwegian army marching to do battle and willing to fight and die for a piece of land that is useless. Hamlet feels disgusted with himself because Fortinbras and his army are willing to fight and die for something so unimportant, when he himself has a father to revenge, but has not yet acted. Again, Fortinbras offers a contrast to Hamlet.
The enotes Study Guide on the play adds the following:
Fortinbras: Fortinbras is the heir to the throne of Norway. His situation resembles that of Hamlet: his father was king, and his uncle is currently ruling. Prior to the play, the old Norwegian King Fortinbras lost both his life and Norwegian lands in the battle with King Hamlet. Early in the play, young Fortinbras is described as seeking to regain the lost Norwegian land during the period of uncertainty following King Hamlet's death. Negotiations between King Claudius and the current king of Norway, however, result in Fortinbras agreeing to cease hostilities in Denmark. He petitions for safe passage through Denmark to Poland. Hamlet describes Fortinbras as "a delicate and tender prince" (IV.iv.48) who is easily incited to fight in the cause of personal or national pride. He passes through Denmark on his return from his conquest of Poland, and is named by the dying Prince Hamlet as the most likely successor to the throne of Denmark. Fortinbras orders a soldier's funeral for Hamlet, and speaks the last words of the play, commending Hamlet as likely to have been a good ruler.