You didn't say which version or what product you have. Generally with Premier Elements you "deregister" using a choice on the help menu for the "old" computer. Version 13 seems a little different in that the Help Menu choice is "Sign Out". Then you can use the computer operating system to uninstall. If you don't deregister or sign out, you will have trouble on the new computer entering the serial number. In some cases, people have to get Adobe to reset your account like for a computer crash.
You are allowed two installations for each serial number you buy. If you've only installed it once, on one computer, you won't have difficulty on the new one. The licensed agreement says both computers have to be yours and you can only use one at a time.
What version of Premiere Elements is involved? And, what computer operating system?
Strictly from the point of view of removing a version no longer wanted on a computer, Deactivate and then Uninstall which are not the same
a. In an opened project, Help Menu/Deactivate or Help Menu/Sign Out (depends on version)
b. Then uninstalling from the computer the usual Control Panel way (assumes Windows).
Please also refer to
Adobe allows you to use the same purchased activation serial number for a given Premiere Elements version on no more than two of your computers.
And, written here by others, not both at the same time.
You can uninstall it on Windows Machine from Control Panel > Program & Features > Uninstall a Program
Go to Applications > Premiere Elements 13 > Uninstall Premiere Elements 13