Ubuntu, being a Debian derivative, is solidy built on APT.
apt-get, the Ubuntu Software Centre and PPAs all depend on APT to work correctly.
Having said that, you can install programs from source, bypassing APT. Have a look at this question for how to do it: How do I install a .tar.gz (or .tar.bz2) file?
In your case, I would not recommend this. I would concentrate on trying to fix the broken catalogue. APT is much superior and easier to use than installing directly from source packages, and you are much more likely to have success fixing APT's problems than you are fixing problems with source installations.
See this question for a summary of the differences between APT and installing straight from source: Should I install programs from a source tarball (`.tar.gz`), from the Ubuntu Software Centre, or from elsewhere?