We don't know what happens in this play past Fortinbras' final words, "Go, bid the soldiers shoot," which is usually viewed as a salute to the dead Prince Hamlet. Since he does have Hamlet's blessing to take the throne and since, as he says, he has some claims to the kingdom of Denmark, then it is a safe assumption that he does become king. Fortinbras is probably not attacking Denmark so much as he is coming in determination. He is determined to avenge his father's death at the hands of Hamlet's father, King Hamlet. He has met up with and is accompanied by ambassadors from England and he has his soldiers salute them with a volley of gunfire hence Osric's words around lines 356-58 of Act 5, sc. 2. Fortinbras does bring soldiers and he does seem to take over immediately, but it's not clear if he invaded in a war-like manner.