On Debian derived systems you can get the source (even as a non root user) using apt-get source
provided sources.list has some deb-src entries. Otherwise it's stuck with downloading from the project source repository with standard tools.
Once you've got that source then:
./configure --prefix=/home/me/mysoftware
make
make install
Will work for most source distributions using autotools or similar.
Once you've done that you'll want to set LD_LIBRARY_PATH and PATH environment variables in your profile.
When you configure other things that depend on software installed in this way you'll need to set PKG_CONFIG_PATH, CFLAGS, CXXFLAGS, LDFLAGS and possibly use other configure arguments to "help" configure find your special install path.
Of course it might be easier just to open a support ticket for someone with the power of root.
sudo apt-get install package-dev
; how do I install them "locally"? – user541686 Aug 14 '14 at 07:54apt-get
. – user541686 Aug 14 '14 at 08:47