Middle Earth Shadow of Mordor: How To Upgrade Sword will show you some of the best ways to level up your weapons in the game.
The Fell Beast – This one is found in the Barrow of Udun and you will need the Ride Caragors which will be the first upgrade to start it. Dominate the beast and ride to the first Guard. To finish the bonus, knock an enemy down first then bite them with the Caragor. Catch all the bad guys with the beast.
Cutting The Lines – Kill some Uruk recruits in this mission and the bonus is to execute 5 of them
Arise – Combat Drain skill to do this one. All you have to do is fight off an Ambush of 40 Uruks.
Nameless Things – Clear 3 Ghul nests and the bonus is to kill 10 with a single Wraith Flash. Build up your combo and do the Wraith Flash to complete the bonus.
Into The Pit – Kill 50 enemies and get 100 combo streak.