Talk to Amdir. Wait for him to kill a guard, then go through the gate and take a weapon from a chest (right-click it to do so). Equip it and talk to Amdir once more. Go through the door that he pointed. You are to defeat 4 guards here - go forth and fight with them. (remember to search the corpses) Eventually you'll meet Celandine Brandybuck. Talk to her and she'll open the next door for you. Go through it and defeat one more enemy, Edrick. Talk to Mundo Sackville-Baggins. Follow the hobbits and watch the scene, where Amdir confronts a Nazgul. When the monster leaves, talk to Amdir, then speak with Celandine. Use the "Finish now" button to complete this mission.