When you installed the license it should have opened a page that contained your license key, which was hyperlinked to a MJP file. I believe they also email this file as an attachment to you so it should be in your email as well. Simply transfer the MJP file to the other computer, make sure MC is installed on it, and double click the MJP file. It will launch MC and install the license.
You can use
my Settings Saving Script to move your settings from the one computer to the other.
One small caveat is that the MJP files only last 2 weeks, afterwhich they "expire". This isn't a huge deal though because you can generate 10 of them per rolling calendar year. In MC on the unregistered machine simply go to the Restoration Page (you can also do this from within MC under the Help menu) and generate a new one. The restore page is here:
https://rover.jriver.com/cgi-bin/restore.cgiI use multiple machines (I think I have MC on 5 different machines at last count) and I reinstall Windows regularly (and upgrade hardware). I've never had a problem with using up my 10 restores, because it's a rolling year (meaning that restore you used on October 10, 2005 will "drop off" tomorrow and become available again).
I love that JRiver has the very respectful policy of allowing a single user to install MC on all of their machines for a single price. I'm quite enthusiastic about it actually...