You should back up your system before installation; you can use Time Machine.

Do not interrupt the installation process once you have started to update your system. You may experience unexpected results if you have third-party system software modifications installed, or if you've modified the operating system through other means.

To update to Mac OS X v10.5.8, use Software Update or the standalone installer.

Software Update

Choose Software Update from the Apple menu to automatically check for the latest Apple software via the Internet, including this update. If your computer is not up-to-date, other software updates available for your computer may appear, which you should install. Note that an update's size may vary from computer to computer when installed using Software Update. Also, some updates must be installed prior to others, so you should run Software Update more than once to make sure you have all available updates.

Standalone installer

Download the update installer and run it manually. This is a useful option when you need to update multiple computers but only want to download the update once. These versions of the standalone installers are available from Apple Support Downloads.