How do I reinstall Photoshop CS6 on my new computer when, upon inputting my KEY I continually get the message "you are not connected to the internet"?

All you need to do is remove the Adobe entries from Hosts file which are blocking activation process.

Steps :

Windows 7 :

Simply run Microsoft Fix it tool to reset Hosts file  to default :

http://go.microsoft.com/?linkid=9668866

Windows 8 :

1. Open Notepad. To do this, swipe in from the right edge of the screen, tap Search, type Notepad, and then tap the Notepad icon. Or, if you are using a mouse, point to the upper-right corner of the screen, move the mouse pointer down, click Search, type Notepad, and then click Notepad.
2. Copy the following text, and then paste the text into the file:
   
   # Copyright (c) 1993-2006 Microsoft Corp. # # This is a sample HOSTS file used by Microsoft TCP/IP for Windows. # # This file contains the mappings of IP addresses to host names. Each # entry should be kept on an individual line. The IP address should # be placed in the first column followed by the corresponding host name. # The IP address and the host name should be separated by at least one # space. # # Additionally, comments (such as these) may be inserted on individual # lines or following the machine name denoted by a '#' symbol. # # For example: # # 102.54.94.97 rhino.acme.com # source server # 38.25.63.10 x.acme.com # x client host # localhost name resolution is handle within DNS itself. # 127.0.0.1 localhost # ::1 localhost
3. On the File menu, tap or click Save as, type "hosts" in the File name box (as displayed in the following image), and then save the file to the desktop.
   
4. Close Notepad.
5. Open the %WinDir%\System32\Drivers\Etc folder. To do this, follow these steps:
   
   1. Swipe in from the right edge of the screen, tap Search, type run, and then tap the Run icon. Or, if you are using a mouse, point to the upper-right corner of the screen, move the mouse pointer down, click Search, type run, and then click the Run icon.
   2. Type %WinDir%\System32\Drivers\Etc in the Open box, and then tap or click OK.
6. Select the Hosts file, tap or click Rename, and then rename the file as "Hosts.old".
7. Copy or move the Hosts file that you created in step 3 to the %WinDir%\System32\Drivers\Etc folder. If you are prompted to enter an administrator password, tap or click Continue.

Once the Hosts file is reset to default , Enter serial number again and try to install CS 6 ..