Neither Claudius nor Fortinbras' uncle want Fortinbras to take revenge on Denmark.
Here's the score:
- King Hamlet kills Fortinbras' father.
- Fortinbras wants revenge on King Hamlet and Denmark.
- Claudius does Fortinbras' dirty work for him and kills King Hamlet but doesn't want Fortinbras to take revenge on him.
- Hamlet wants revenge on Claudius for killing his father.
- Fortinbras' uncle, through Claudius' urging, convinces Fortinbras not to take revenge on Denmark.
- Claudius ends up doing Fortinbras' dirty work for him by killing Hamlet, the Queen, himself, and leaving Denmark open for invasion.
Now the long version:
Fortinbras had wanted to enact revenge on Denmark through war; he saw Claudius as weak, compared to King Hamlet. As for Fortinbras' uncle, he is sick, too weak to be a threat to Claudius. Claudius fears Fortinbras, so he writes the sick uncle not to let Fortinbras take revenge on Denmark.
Instead, Claudius wants revenge on Hamlet for trying to take revenge on him. It's incestuous revenge that is the problem in this play. Claudius' political plan works: Fortinbras' uncle convinces Fortinbras to thwart his revenge. Tragically, no one intervenes on behalf of Hamlet's plan of revenge.
Fortinbras and Hamlet are foils. Both of their father's (the Kings) have been murdered, and both of them (the Princes) have been passed over for the crown by their uncles. Both have legitimate claims to the throne. Both have rights to enact revenge. But Fortinbras does not for two reasons: 1) he was not visited by his father's ghost who demanded revenge; 2) his uncle interceded for him.
Fortinbras is more practical than Hamlet. Instead of hastily enacting a ill-conceived personal plan of revenge against a well-guarded king, Fortinbras waits. He says:
I have some rights of memory in this kingdom,
Which now to claim my vantage doth invite me.
Fortinbras bides his time, unlike the melancholy Dane. Soon enough, his time does come, and he not only gets the title promised him, but he marches into Denmark, which is in complete disarray, and claims it too! His claim is even greater than he ever could have hoped. He even gets Hamlet's claim:
But I do prophesy th' election lights
On Fortinbras. He has my dying voice.