Upgrading from request-tracker 3.8.2 to 3.8.3 is a relatively easy procedure.

Become root:
sudo -s

Download request-tracker 3.8.3:
wget http://download.bestpractical.com/pub/rt/release/rt-3.8.3.tar.gz

Untar the archive:
tar zxvf rt-3.8.3.tar.gz
cd rt-3.8.3

Configure:
./configure

Test your dependencies:
make testdeps

All dependencies should be found. If not, you may have to download and install these through CPAN. See this blog post for more information on satisfying request-tracker dependencies.

Perform the upgrade:
make upgrade

When asked what version you are upgrading from, enter 3.8.2

Upgrade the database:
/opt/rt3/sbin/rt-setup-database --dba root --prompt-for-dba-password --action upgrade

Restart apache:
/etc/init.d/apache2 restart

The upgrade process has completed. Browse to your request tracker installation, which will now display 3.8.3 at the logon screen.