There are two doors there, one is locked and you cannot go through there yet. Go through the open door which should be marked with the quest waypoint. There is a East Empire Company Strongbox close to the entrance with an expert difficulty lock. Keep going down, until you arrive at a chamber that has a Locked Wooden Door. The key is not in this room; find a passageway leading you deeper into the fort.
Arrive at another big room with a wooden coffin and a Knapsack laying against the coffin. Loot the Knapsack to find the key to the wooden door, the Fort Frostmoth Key. Go back up to the locked door and unlock it using the key.
Soon you will get to General Falx Carius's Quarters in which you will fight General Falx Carius. After you defeat him and his Spawns, grab the unique mace, Champion's Cudgel, as well as another Fort Frostmoth Key from his body. Also be sure to open the chest in the room. On occasion, it will have an armor made of Stalhrim such as a Stalhrim Helmet of Extreme Illusion.
Finally, leave the room through the back door to exit back to Solstheim