I have got some issues when installing the Debian package for Vims Latex Suite. Finally, it is working now. Using the Vim Addon Manager instead of adding the plugin manually was the solution.

Of course, you have to install the Debian packages first (as root):
# aptitude install vim-latexsuite vim-addon-manager

Now you can check everything went its way:
$ vim-addons status
This creates the following output:

# Name User Status System Status
…
latex-suite removed removed
…

This indicates that you have to install the plugin:
vim-addons install latex-suite

A short check shows the success of the action:
vim-addons status

…
latex-suite installed removed
…