Exactly the same way you did the first time... how DID you install on the old computer?
If you are not going to use the old computer again, be sure to deactivate and uninstall
Did you purchase Premiere Elements 12 as an online download from Adobe? And, do you have a purchased serial number for it?
If you have the purchased serial number and purchased online download from Adobe, you should be able to go to http://www.adobe.com and "My Orders" to download another set of installation files. Quickest way to get to "My Orders" - go to http://www.adobe.com, sign in with your Adobe ID and password, type in "My Order" in the Search field at the top right of the Adobe web page.
But, there are some other details to be considered...
What is the operating system of the old computer to which you downloaded your purchased Premiere Elements 12? Also, 32 or 64 bit old computer?
What is the operating system of the new computer where you want Premiere Elements 12 installed? Also, 32 or 64 bit new computer?
Do you have Premiere Elements 12 now installed and activated on only the old computer?
The operating system of the computer to which the purchase was download and the number of computers with the installed activated same serial number are important.
We will be watching for more information.
Just an add on to my post...
Doing exactly what you did for old may or may not be applicable to the new.
We need the information requested to advise you so that you do not run into any surprises as you go to perform the task.