Mon, 30 Sep 2013
At this time, the latest Linux kernel that has been specially prepared for use with Ubuntu 12.04 is based on Linux 3.8. The rest of this answer is about how to get Linux 3.8 within Ubuntu 12.04.
(Before I get into that, first an aside: you've linked to the "mainline kernel" PPA, a collection of Linux kernel packages that have not been specially tested and prepared by the Ubuntu kernel team. You can read more about mainline kernels here. The mainline kernels are not suggested for installation, except if you are doing tests or you experience a major problem that it can be resolved with a specific mainline kernel.)
The recommended way to get the latest kernel on Ubuntu 12.04 is to stay within Ubuntu 12.04, rather than enabling separate package archives like the PPA you linked to. The latest Ubuntu 12.04 provides special package names to install the more up-to-date packages.
To install them, open a terminal (CTRL+ALT+T) and issue the following commands.
sudo apt-get install linux-image-generic-lts-raring sudo apt-get install linux-headers-generic-lts-raring
When the installation completes, you can reboot your system to boot from 3.8 kernel. (The above commands won't uninstall anything (unless your APT is somehow misconfigured), and definitely won't auto-reboot, so you can run the commands and then do the reboot at some later time.)
The bootloader that Ubuntu uses (Grub) lists the newer kernel first, so it will boot from 3.8 automatically (except if you have installed a newer than 3.8) (and except if you know you've customized Grub somehow).
A note about why 3.8: Because of the Long Term Support status of Ubuntu 12.04, many different kernels will be backported (aka specially prepared) from newer versions of Ubuntu. (Additionally, a backported graphics stack (X and related packages) is available.) The 12.04.3 version (that has been released on 22 August 2013) contains the latest kernel and graphics (X) packages from Ubuntu 13.04 (raring). Ubuntu raring has 3.8, so therefore the "backported" version of Linux from raring to precise is also 3.8.
A note about the best way to get an even fresher version, like 3.11: Wait for Ubuntu Saucy (13.10) to come out, and then wait (not very long) for the Ubuntu Long Term Support process to backport that to Precise (12.04). Per the release schedule, Saucy should be out on October 17. Then you will be able to install linux-image-generic-lts-saucy!