to check the firmware, open your router's configuration page typically
http://192.168.1.1 in your internet explorer. You may be asked for a user name and password the default is "admin" "admin".
The current firmware version will be displayed in either in the top right of the screen, or by looking at the status or admin tabs.
You can find the current firmware on the Linksys.co.uk website under "support". There are a number of versions of the Linksys router so check it's underside, or front to see whether you have version 1, 2, 3 etc (version 1 will probably not show a version number, just the model).
Download the new firmware to your computer and unzip it.
To update the firmware (only do this on a wired, not wireless connection), look at the administration tab of the router's control panel screen, find the firmware options and browse to the file you just downloaded from linksys. The router should install - don't touch it while it does, and then reboot (your screen may disconnect or display other messages).