Illarion.org

Is it possible to log onto your character using another computer with Illarion installed on to it?

Yes, it is.
If you copy your names.dat on a floppy and copy it on the other computer, you even know the people that already introduced to you.

So it's just the same as running it from the origional computer??

Yes. All character informations except the people you know and the map you explored are saved on the gameserver.