Quick and dirty
Install maven on the linux server and a JDK. Then copy the source code of your project to the linux server (e.g. clone the source repository or zip the directory and scp it). Run:
mvn clean test
A bit more complex but nicer
Using the Selenium Grid. The Grid allows selenium to start a browser from remote nodes.
First download the selenium server from the Google Code download page. Start the hub on your windows machine:
java -jar selenium-server-standalone-2.39.0.jar -role hub
Go to http://localhost:4444/grid/console
to check that the hub has correctly started.
Then on the linux server start a node:
java -jar selenium-server-standalone-2.7.0.jar -role webdriver -hub http://<hub_ip_or_hostname>:4444/grid/register -port 5556 -browser browserName=firefox
In your test code, instantiate a RemoteWebDriver
object with the Firefox capability. The remote webdriver object will automatically contact the hub to find a remote node with the requested capabilities (and here there is only one single node). Then the hub will forward the selenium commands to the remote node.
DesiredCapabilities capability = DesiredCapabilities.firefox();
WebDriver driver = new RemoteWebDriver(new URL("http://localhost:4444/wd/hub"), capability);
driver.go("http://www.myWebsiteToBeTested/");
Then you can add to the hub as many nodes as you wish, using different platforms and browsers (Firefox on Linux, IE on Windows, Safari on iOS...). The official documentation is here.