In this interview, Mark Shuttleworth mentions Arkham kernels. I have tried finding these with Google searches, searching launchpad.net, searching ubuntu.com, but I have not been able to find any further information.
Where can I find further information on Arkham, such as how can I install an Arkham kernel?
An Arkham kernel is not something you can install. Arkham appears to refer to a set of kernel patches in source code form. Google does more or less the same thing for its Android kernels.
The purpose is to have a set of patches specific for Ubuntu (including AppArmor/snap/etc. features upstream kernel developers are not yet prepared to accept, as well as HWE i.e. backports of hardware support features from newer kernels). This makes it easier for Ubuntu developers to just
git rebase
the whole set of patches from one kernel version to a later version so that they can develop faster, and also have the whole set tested together.To find a set of Arkham kernel patches, you can go to something like https://git.launchpad.net/~ubuntu-kernel/ubuntu/+source/linux/+git/mantic/log/?h=master-prep (Markdown broke the link. Copy the link destination and replace %7E with a tilde). You can also use the distribution you are interested in instead of mantic. Then all the commits that have a committer or signoff with "@canonical.com" are part of the set of patches known as the "Arkham kernel."
You might still want to install it. Good news. The Ubuntu kernel you have installed by default already includes these code changes.