My recommendation is that you don't do this and instead use a precompiled version.
However, if that's not good enough:
Is it right to run the ./configure on the raspberry?
Then copy the node folder to my desktop computer and run make.
No. The build must be done natively on the pi, or else with the use of a cross compiler. In the former case, resolving dependencies will be easy since you can just install them w/
apt-get, but the build will be SLOW. In the latter case, the build will probably be much much faster (presuming your desktop is not also a 700 Mhz single core), but getting it set up and resolving dependencies in the cross environment will probably take you a lot of time to learn and is more prone to error -- including the possibility of subtle flaws in the executable.
The default install directory is probably
/usr/local, so you should do that step as root (I don't know how well
sudo make ... works), or else set
./configure --help should provide some information about this.