First, try apt-get with the version you'd like. For instance (and I haven't looked if 2.8.0 is even in the default 10.4 Ubuntu repos)
$ sudo apt-get install libxml2=2.8.0
but use that with caution as it can cause dependency downgrades and APT problems. You have the source, and that's usually a better place to be.
The source package is built around GNU autotools
$ ./configure
$ make
$ make check
$# if all is ok, then
$ sudo make install
For options and more control over features, start with
$ ./configure --help
and look to see if you'd need things like --with-run-debug etcetera.
For autotools source builds, you may have to dig around for the development headers required for some dependencies. ./configure usually informs you of what is missing, then go look for -dev versions of the packages. You won't need it, as you have the source tree, but as an example apt-get install libxml2-dev
The default installation top level for apt-get is usually /usr, and for GNU autotools, and source builds, it is usually /usr/local. This is a good thing, but can make looking for libraries (from a human point of view) a little trickier, until you get used to the lay of the land.