Which is the best way to install new hplip versions?

You can use the official hp installer as I did. It will detect that I'm using Ubuntu, uninstall the older drivers, fetch all the dependencies (includind build-essentials), compile the new version, make a nice deb and install it.

Great features:

• It's clean, it uninstalls old versions and install as a deb.
• It's always the latest stable version, it quickly adds support to newer printers.
• It worked with all the Ubuntu versions that I throw at it. Likely it's also cross-distro.

Problems:

• Takes loads of time to install, it must compile the driver and fetch hundreds of megas of dependencies from the internet.
• It doesn't integrate into Ubuntu Update, so no automatically security updates.
• Does it really need to add an HP icon to the tray?.

It's a great piece of software, technically it's amazing, but the first two problems are a issue for me.

Installing from the Official Backports in Ubuntu 12.04 LTS

At least for Ubuntu 12.04 LTS there is a relatively new version of HPLIP available in the official precise-backports repository. See this answer for how to enable the backports repository (should be enabled by default).

You can then install the latest version of HPLIP from the backports repository like so:

sudo apt-get --target-release precise-backports install hplip

IMHO, this should be the “technically” best solution to get a very recent version of HPLIP (at least for Ubuntu 12.04 LTS) as this solution does not pose the problems mentioned in Javier Rivera’s answer.

I am currently running Ubuntu 12.04 (precise), which has version 3.12.2 of hplip. But I found out from the hplip site that I needed at least version 3.12.4 of hplip for my new HP Officejet printer.

I noticed that the next most recent Ubuntu release, 12.10 (quantal), had hplip version 3.12.6, but I didn't want to upgrade my entire system yet just for that. I also preferred to stay closer to a consistent Ubuntu setup than what Javier ended up doing. So here is what I did:

I went to the Ubuntu packages site and downloaded the latest deb file from 12.10 (quantal) for my architecture (amd64) for each of the following packages:

libhpmud0
printer-driver-hpcups
libsane-hpaio
hplip-data
hplip

I then installed each of the deb files, in that order, using this command:

sudo dpkg -i <deb-file-name>

[Actually, first I tried doing that just for hplip. It failed due to missing dependencies. I kept following the dependencies until I compiled that list.]

I think that is all I need, really, but for completeness, I did the same thing for two more packages:

printer-driver-hpijs
hpijs-ppds

Now my HP Officejet printer works perfectly, fully supported by genuine Ubuntu packages.

The only thing left to do is to figure out how to configure the files in /etc/apt to watch for updates for the quantal packages I installed. I'll update this post if and when I get around to doing that, or maybe someone else can help.