While I don't currently have time to provide a super detailed answer, I can outline a possible strategy for running executables from a pre-built git package under your home directory. This response is aimed at Debian, since that's what you said you're using, but the basic concept is applicable to other OSes and distros as well.
First, go to http://packages.debian.org/squeeze/git and review the dependencies of the git package. Luckily, you'll find that git doesn't have that many. Most of the important ones are probably already installed, maybe with the exception of libcurl3-gnutls and libexpat1.
Next, download the binary .deb package and extract its contents (search the web to find out how to extract a .deb package manually). You'll find a data.tar.gz file. Extract that, and you'll get usr/bin/git* and usr/lib/git-core/* files. You can put these files somewhere under your home directory. Now the critical part: /usr/bin/git calls out to numerous "helper" programs under /usr/lib/git-core, and this latter path is hardcoded into /usr/bin/git. Luckily, you can override the hardcoded value by setting the GIT_EXEC_PATH environment variable, so update your dotfiles to point to where you decided to keep the various /usr/lib/git-core files.
Finally, it's possible that not all of the library dependencies (and their dependencies, etc, etc) were installed on your system (e.g., libcurl3-gnutls, libexpat1). You can download the binary packages for these as well, and extract them under your home directory, and then help git find them by setting your LD_LIBRARY_PATH environment variable appropriately.
P.S. I've probably had more than my fair share of being in your position, so what I've outlined above is more than theoretical. But working around this sort of problem is a little bit of a black art, so there will usually be some additional fussing around involved. Good luck.