I have upgraded my diskless PXE booting systems in the past using do-release-upgrade.
Install update-manager-core to obtain do-release-upgrade. When you upgrade, don't forget to ensure that your initrd is built for netbooting in /etc/initramfs-tools/initramfs.conf, I have had this file overwritten on upgrade in the past, the option you need is:
You will also need to update your kernel and kernel configuration on your tftp server.
Edit the configuration file in your pxelinux.cfg directory to contain the new entry, on my server I have a host specific file:
johnf@carbon:~$ cat /srv/tftp/pxelinux.cfg/01-00-1f-d0-27-d2-17
APPEND initrd=/initrd.img-2.6.38-10-generic root=/dev/nfs nfsroot=172.29.1.6:/home/export/argon ip=dhcp rw
You will also need to update the kernel images on the tftp server itself, here's the command I use:
sudo cp --preserve=timestamps /home/export/argon/boot/initrd.img*-generic /home/export/argon/boot/vmlinuz*-generic /srv/tftp