I don't know why you'd want a second copy of City on your computer, unless it's meant to be for use with the Test Server. In any event, to have a copy of your installation, make a new folder in the location of your choice (not in the City of Heroes folder) and copy your City of Heroes folder to it.
If you were trying to make a test server installation, rename the new copy of "City of Heroes" to "cohtest"
Trying to be more helpful and coherent...
It's OK to copy your City of Heroes folder to the same location, so that you have "City of Heroes" and "Copy of City of Heroes" next to it. Rename the "Copy of..." folder cohtest after you finish copying.
Find CohUpdater.exe somewhere other than in the cohtest folder. There should be one in your City of Heroes folder. Right click on it and select Create Shortcut form the context menu. You can move this new shortcut to your desktop, and rename it something helpful like Training Room.
Right-click the shortcut and select Properties. In the Target window, add a space and -test to the end of the string that's there. It will probably have quotes around it; make your addition outside the quotes. It will probably look something like
"C:\Program Files\City of Heroes\CohUpdater.exe" -test
Click OK.
Double click your new shortcut. You should get a dialog asking you to verify where the client will be installed. Make sure it is the folder that contains your new cohtest folder. Browse to that folder if required.
When the folder name is correct in the dialog, click yes. The updater should now run normally and apply any necessary patches. It won't download the whole game.
I tried to get on Test this morning. Either the update server is down, or they're testing something supa secret that requires a special updater. I just get Connecting... #23... #24... so this won't work right now, unless you've gotten an invitation that I haven't.
The updater for Test is currently mostly broken.
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I just installed a second copy of coh to my computer and am having some difficulties. But before I ask, when installing the second copy should I install to the same folder as my regular coh? I ask this because when installing my computer asks me wether or not I want to install it in the same folder or not. When my computer asks me this it says if I do not install to the same folder, there will be problems with uninstalling. I am curious as to if this would be true. Also since I installed a second copy of coh shouldn't there be second copies of the program files in the coh folder if I installed to the same folder (asking because there is only one copy still, Also if it is in the same folder would it have just instaled over the origional, deleting previous patches.) A side note, in my start menu there are two files one with regular coh in it the other with test, if I did the test shortcut on the second start menu file would that allow for differentiation between the coh files? I'm asking this because I am really hoping I did not make a mistake and have to uninstall and re-install it all. (sorry for wall of text, tab button busted)