What OS did your previous computer have, if Windows 8 is like Windows 7, it's a pain in the butt to set up, but once you know where to go, its pretty easy. With Windows 7 you had to go online as an "Administrator" and go to the CareLink site to set up the account.
Do you mean the Internet Explorer or any other browser must be started as an "Administrator"? If I understood this correctly then these are the next steps:
Right click on the icon of the browser on the desktop (I am not sure if carelink will run in the Metro IE). In the menu that will appear select "Run as Administrator" and then visit the carelink site. This process will allow Java to make all the necessary changes to install the carelink software on your computer. As a prerequisite the latest Java should be installed.
I assume just Internet Explores. I'm not sure if you use another browser. And thanks for including all the steps, It's been a while since I've messed with Windows, I've switched to Mac. I just remembered it being a HUGE pain to set up on Windows 7.