I recently added the Docker extension onto VS code. Then I downloaded Docker, after installation I received this exception:
.NET Framework Error- Unhandled exception- exception type 'Docker.ApiServices.WSL2.WslKernel`
I'm not quite sure where to proceed from here.

I had the same exception and command:
wsl --set-default-version 2
returned message:
An update to the WSL 2 kernel component is required. For information, visit the https://aka.ms/wsl2kernel website. (This is not original message, I've translated it from my native language.)
When I've followed link and installed kernel update docker starts working.
I believe Docker is trying to run on Windows WSL framework/feature. Please make sure you've turned it on.
How to do that?

wsl --set-default-version 2, this will make your system switch to latest version of WSL with added benefits.If this answer helps, let me know, or I can go in further details.