Approach the fort. As you climb towards the main gate, you might get attacked by Ash Spawn and other enemies. Note the watchtower on top of the hill, there are several enemies there too. As you enter the main fort area, you will hear General Falx Carius calling out to his men to prepare an ambush for the invader (you), and other calls such as "Fort Frostmoth will never fall! Long live the empire!"
Enter the courtyard, and clear any additional enemies there. Then open the door that will lead you to inner Fort Frostmoth. See the map below to help you navigate this dungeon. X marks the spot where General Carius is.
There will be more Ash Spawn inside. After those have been neutralized, check the area near the entrance for some loot including an East Empire Company Strongbox with an expert lock. Then continue deeper into the fort. Go down some stairs. There is a door there with a master lock. Pick it if you can. There is a chest there that contains valuable items, as well as other miscellaneous items spread about in that cell.
Exit the cell and note the wooden door on the other side of the door. It cannot be picked, and requires a key. Search the wall for an exit to a passage leading further down in the fort. Go through some cobwebs, then turn left and head through a corridor to reach a chamber with some sort of wooden coffin. There is a knapsack on the floor next to the coffin; Search the Knapsack to get the Fort Frostmoth Key, as well as other random loot.
Now you can go back up to where the locked wooden door was and use the Fort Frostmoth Key. Go up the stairs behind the doorway, and there you will find the quarters of General Falx Carius. Kill Falx Carius. This can be a challenging fight because he will have many Ash Spawns helping him. If you have a follower you can summon, now would be a good time.
Once you have defeated General Falx Carius, your final objective in this quest is to return to Captain Veleth. Before leaving Falx Carius's room, make sure to search the room for valuable loot. General Carius himself drops the Champion's Cudgel, a unique rare two-handed mace. Also loot Falx's body for another Fort Frostmoth Key. There is also a chest here, which often contains a piece of Stalhrim Armor, such as the Stalhrim Helmet of Extreme Illusion.
You don't have to go all the way back, as their is a door behind where you can exit directly to Solstheim.
Note that Captain Veleth is no longer at the Attius Farm - he has returned to Raven Rock. So Fast Travel to Raven Rock and talk to Captain Veleth. He is usually moving around, use the quest marker on your compass to locate him. Or he could be asleep in the Bulwark, but you can just wake him up. Veleth asks you what news you bring from Fort Frostmoth. Tell him you killed General Carius. Veleth suspected that Carius was undead, how else could he have survived for over 200 years? It's a shame. There are quite a few tales of General Carius's exploits, including the founding of Raven Rock.
Assure Captain Veleth that Carius will no longer send ash spawn to Raven Rock. Veleth tells you that he spoke with Councilor Morvayn who wanted him to give something to you if you came back in one piece and got rid of General Carius. Better than a soldier's pay, so you should be thankful.
Captain Veleth gives you 2500 gold as a reward. This completes the quest, March of the Dead.