How do you get to Northwatch Keep?

I find the Clairvoyance spell quite handy for finding new locations, especially when mountains are involved. Just make sure that your current quest is the only active quest, and that you do not have a custom destination set. Then go to the quest and do "Show On Map", and fast travel to the closest destination you've already visited. You can then repeatedly use Clairvoyance to show you the way.

It likes to take roads/paths as much as possible so it may not be the most direct route, but should get you there eventually.

I would say that the best route to get to Northwatch Keep is by following the shore from either the West or East.

If you are looking for more specific directions, take a look at this guide.

The fastest way to get there (if you have not been to Solitude already) is to hire the carriage near the Whiterun Stables to take you to Solitude, then from Solitude head west towards the keep. When you get to Steepfall Burrow, turn north and descend to the shoreline. Then keep going west and you will reach Avulstein and his men who are waiting outside the keep.

I had trouble with this too. Here's what I did:

If you head from Dragonbridge towards Rimerock/Lost Echo Cave, you know there's a small fork in the road not far from there. Left will take you to Rimerock and Lost Echo. Instead, go right. (Just past the fork on that side is a small alcove-type area that will unlock Wolfskull Cave.) Follow the path almost all the way down until you can access the shoreline. There are some wolves that will attack on the way. When you run into a farmer and his wife asking for help, you're getting close to the shoreline. (They tell you to watch out for dragons, but I didn't see any.) Walk off the path and down to the water.

Stay as close to the water's edge as possible and avoid or kill any enemies. You should come across a Vigilant of Stendorr fighting some crabs. Keep on for a while and you'll see a wrecked boat in the water. Just past that is a walrus (I didn't stick around long enough to see if it was friendly, so I'm not sure if they'll harm you or not.) and then Lower Steepfall Burrow. You should pass some more walruses shortly after that, and then you're nearly there. Just keep going until you see Avulstein (if you agreed to join him) or the marker.

I got there after getting lost in the woods near Lost Echo Cave. After heading down the path past the cave looking for a way down the mountain and only getting higher into the mountain, I finally trudged back up the path to the rocks overlooking the shoreline below, went a little right, and found a way to jump down the rock to the shore. You land just to the right of the seals.

I kept left, away from the seals, keeping close to the rocks on my left. As you round the bend, you are suddenly there, behind rocks, looking at the keep, with the guy you are supposed to meet.