I have been given the test of strength by the Skaal and have spoken to their shaman by the lake.
I swam down to the cave entrance and ran around all the corridors knocking dragrs down all over the place.
But for the life of me I can't find Aesliip!
On the map I can see a corridor type thing to the right of me when I first enter the caves, but I can't find a way into them.
In desparation, I even looked up a walkthrough which says to get to Aesliip you go through a corridor to the south and east of the cave entrance but I can't even find that! I thought maybe I had killed him in my indiscriminate draugr-bashing session but I didn't find a ring on any of the bodies.
Obviously I'm missing something here but I don't know what it is.

Can anyone help me out??

The caves are somewhat confusing but you are unlikely to have killed Aesliip. You have to change cells to get to his lair, not something it's easy to miss. You have obviously not tried every possibility. Mapping the cave yourself should let you know where you've missed a turning. Nothing that is shown on the cave map is inaccessible from the tunnels. Drop an object at each turn if that helps.

Or post a note. (Double-click on the map, type something like "South-Dead end"). If you do drop stuff, you might want to use lights, so you can easily see them.

Thanks peoples... but I actually figured it out not long after I posted the message (yes that sounds dumb but I was actually running round that same cave for about half an hour!)