The TeamCity installation package (Teamcity-<version number>.tar.gz) can be downloaded from the download page of http://www.jetbrains.com/teamcity/download/.
TeamCity needs JRE or JDK Version 1.6+ to be installed. It is ideal to have the JDK instead of the agent.
Note
The installation package is bundled with the Tomcat 7 servlet container. This is the recommended way to install TeamCity on Linux, unless you want to absolutely use your own installation of Tomcat or a different J2EE servlet container.
We will be using Ubuntu 12.04 for the following steps. It should be straightforward to adapt it to the distribution of your choice.