How are Old Hamlet and Claudius different when it comes to foreign policy?
We learn quite a bit about this topic in the first two scenes of the play. When the soldiers are out on the platform of the castle to look out for the ghost, they also ask Horatio about the build up of armaments and ships that is currently going on. Horatio reveals that Young Fortinbras of Norway is planning on attacking Denmark to gain back lands lost by his father in a battle with the late King Hamlet. This reveals that the late king was a successful warrior who settled land disputes with war. King Hamlet engaged in a battle and "by a sealed compact, well ratified by law and heraldry / Did forfeit, with his life, all those his lands / Which he stood seized of, to the conquerer."
On the other hand, Claudius isn't jumping into a battle with Prince Fortinbras to defend the lands. While is he preparing for the possibility, he is trying diplomacy first. In his first monologue to his court he explains what this external threat to Denmark is, but then tells them that he plans to send a letter to the King of Norway (Fortinbras's bed-ridden old uncle) telling the king that he must stop Young Fortinbras in his plans against Denmark. In a effort at confidence, or in a moment of ego, he is speaking dismissively of the Fortinbras threat and ends that part of his speech with the phrase, "so much for him" that seems to even imply a dismissive wave of the hand to go along with it.
We learn in Act 2 the letter worked and that Old Norway has attempted to reign in his nephew, but who now is only asking to be allowed to pass through Denmark on their way to battle in Poland. And while we may question the honesty of the request, we have to take it at face value for the next couple of acts.