If you have to install software manually from source, make sure to read its installation instructions carefully, in particular, look for a section about user-specific or local installation which describes the way to install the software in your HOME directory.

If the installation follows the common configure - make - make install regime, you can easily change the installation location with the --prefix option to the configure step:

$> ./configure --prefix=$HOME

Also for other installation regimes there is often a --prefix option or something similar (e.g. --user, --installpath).

You can find more information in the documentation about compiling.