I myself just upgraded to KDE 3.1.3 so here is what I did:
1. at the command line as root, type:
$ kedit /etc/apt/sources.list
2. Save a back-up copy, then delete everything in the file. Copy the following line and save:
deb
http://http.us.debian.org/debian unstable main
(note, you might want a closer server than http.us.debian.org.debian, such as
deb
ftp://ftp.it.debian.org/debian unstable main
3. At the command line as root type:
$ apt-get update
4. You can now start the copying process by typing:
$ apt-get install xxxx
where xxxx are the following files:
arts
kde-i18n-it <-- Only if you want Italian modifications to KDE
kdeaddons
kdeadmin
kdeartwork
kdebase
kdebindings
kdeedu
kdegames
kdegraphics
kdelibs
kdemultimedia
kdenetwork
kdepim
kdetoys
kdeutils
quanta
5. Restart your computer and you should be done!