LiveJournal in the past has been server API agnostic, working on either FastCGI or mod_perl, but we've dropped FastCGI support, as there's really no need to maintain it anymore. LiveJournal now solely requires mod_perl; seeing as how it's everywhere and well supported, and you might already have it on your machine. If not, you can read up on the official installation docs located at http://perl.apache.org/docs/1.0/guide/install.html
We've found that more people have success when mod_perl is statically linked, rather than using a DSO. 
Typically, if you already have a working apache installation in Debian and would like to install mod_perl support, then installing the package libapache-mod-perl should suit your needs. Otherwise, you should install the package apache-perl.
If you want the LiveJournal code to run as a user other than what your webserver runs as, then you'll need suEXEC support in Apache. You're free to figure that out yourself. We'll assume your webserver is running as user lj, though it doesn't have to run as any certain user to work.
In fact, the LiveJournal code never writes to disk, so as long as it can write to the database (which is essential, obviously), you can run the code as any user, including nobody or www-data, both of which are common in out-of-the-box configurations.
You can run the LiveJournal code inside or outside of a <VirtualHost> section in Apache's httpd.conf file. However, make note that you can't (yet) have two VirtualHosts running parallel copies of the code. In the future we'll fix that, but for now if you really need two installations on the same machine, you'll need to run two sets of Apache processes, listening on different ports.
 Consult “ mod_perl Installation User Guide: When DSO Can Be Used” for more information.