By default (Using all repos) 13.04 has Chromium version: 28.0.1500.52-0ubuntu1.13.04.3
I found the following while searching the PPA list in Launchpad:
Chromium Beta Builds - https://launchpad.net/~saiarcot895/+archive/chromium-beta (chromium-browser version 31.0.1650.34 as of 2013-11-01)
Chromium Dev Builds - https://launchpad.net/~saiarcot895/+archive/chromium-dev (chromium-browser version 32.0.1671.3 as of 2013-11-01)
API Keys
Warning: Google API Keys are not included with these packages as mentioned in both PPA Links. The process of creating them is quite involved and only recommended for developers!
Do not update chromium-browser from those PPAs if you haven't understood the process below!
NOTE: Some Chromium features, including Sync, require an API key,
which is not included with the packages below. To use these features,
please follow the 11 steps at
http://www.chromium.org/developers/how-tos/api-keys. Then, open
.profile located in your home folder (you will need to see hidden
files), and add the following three lines to the end of the file (on a
new line):
export GOOGLE_API_KEY=<api key>
export GOOGLE_DEFAULT_CLIENT_ID=<client id>
export GOOGLE_DEFAULT_CLIENT_SECRET=<client secret>
Replace what is in the angle brackets (after the '=' sign in each
line) with the appropriate value. You will need to log off and log
back on for Chromium to use the keys.
Installation
To add either of them do the following
Adding Beta PPA
sudo add-apt-repository ppa:saiarcot895/chromium-beta
sudo apt-get update
sudo apt-get install chromium-browser
Adding Dev PPA
sudo add-apt-repository ppa:saiarcot895/chromium-dev
sudo apt-get update
sudo apt-get install chromium-browser
If you already have it installed then simply do a sudo apt-get ugprade
instead of installing the package again. Your end result should be something like this:

Now if you want the Trunk version, please see the answer by Radu
UPDATE - Remember to add the API Keys since when opening for the first time Chromium you will get this:

And basically all Google services (Like syncing for example) will not work. So follow the How To above about adding the Api-Keys. To open the profile simply type the following on the terminal:
gedit ~/.profile
And then follow what the instructions say.
Know that this PPAs only have 1 person (The same person) which means this could be a temp solution (Totally depends on the PPA Maintainer).
Firefox is in main
, and Canonical commits to providing security updates for software in main
(which, in the case of a browser, is all of them). Chromium isn't in main
, it is universe
- so it is updated by volunteers. It might have simply become too much of an effort to update Chromium for 12.04, all the more so since Google doesn't support 12.04 for Google Chrome, so breaking changes might have been made since.
For 14.04, the latest versions of Chromium have failed to build, bugs have been filed with no response from maintainers. Since Chrome still supports 14.04, I assume this might be fixable, and no one has volunteered to fix it.
Best Answer
Chromium is currently not built for the chromium-daily PPA, which makes using it a bit of a security issue. Unless you can find a better PPA that stays up to date, I'd recommend you use the repo version of Chromium. Using an old browser on the internet will get you hacked. See:
Does someone know why the Chromium daily package isn't build anymore?
For the latest stable of Chromium click the button here:
https://snapcraft.io/chromium