There are two parts to this question plus an unstated part:
- where should I temporarily save the downloaded tbz2 file and work on it?
- where should I install the executables, the man pages, the config files and so on?
- what other approaches are there?
People sometimes create a subdirectory in /tmp (e.g. /tmp/ffmpeg) and extract the files there. You need to clean up and delete after the software is installed.
I tend to login as a normal user, create a sysadmin/software folder in my home directory and extract new software into subdirectories there. I only su to root when the time comes to install the compiled and tested binaries etc.
There are standards for where software should be installed. Read the Linux Filesystem Hierarchy Standards, http://en.wikipedia.org/wiki/Filesystem_Hierarchy_Standard
I prefer to use the platform's packaging system and software repositories. This depends on which Linux distribution you use. For RedHat/CentOS etc I would not bother with the download unless the repositories had no ffmpeg-php package I could use with
yum install ffmpeg-php. For other distros the native package management tool might be apt-get. There are advantages to having the package manager handle conflicts, dependencies and be able to provide a complete list of installed packages. I've no idea if ffmpeg-php is available this way - so it might not be an option, but I'd check carefully first.