The answer is yes, we can assume that Fortinbras becomes king at the end of the play. As to your second question, there is no evidence to support the fact that Fortinbras was attacking Denmark.
The text tells us that Fortinbras had just come from battles in Poland. There is no indication that he originally intended to attack Denmark, however. We do know that Fortinbras' father (the king of Norway) was slain by Hamlet's father, the king of Denmark, earlier in the play.
It appears that Fortinbras has arrived at an opportune moment in Act 5, Scene 2. In this scene, Fortinbras acknowledges his good fortune but refrains from rejoicing that Hamlet is dead. He does admit, however, that the way is now clear for him to assume the Danish throne. By all indications, Fortinbras takes charge immediately, after realizing that both King Claudius and Hamlet are dead.
Let us haste to hear it,
And call the noblest to the audience.
For me, with sorrow I embrace my fortune.
I have some rights of memory in this kingdom,
Which now to claim my vantage doth invite me.
He orders a military funeral for Hamlet, with all the requisite military rites. Fortinbras also orders that the corpses be dealt with immediately. Then, he tells Horatio to go outside and tell the soldiers to fire their guns in Hamlet's honor. By all indications, Fortinbras acts and speaks as a king. So, we can assume that he did become the king of Denmark after Hamlet's death.