You don't have to install the OS. A Live CD or DVD (or USB key) means it's bootable. You
can install it if you want (I like having a Linux build in addition to Windows), but you can run everything and alter files and even install programs during that boot session. Once you shut the computer down, though, any personalization you've done is wiped, as are any files you saved
in that partition (unless you then install it).
From here on out I'll talk about
Ubuntu, since that's the Linux build I can give the best instructions for.
While you're working in Ubuntu, your Windows file system will appear as a mountable drive. The title varies depending on the computer, but once you're inside it looks just like when you're looking at C:\ in Windows Explorer. So go to that drive, and the Origin Games folder should be right there. You can then navigate in and delete the file, then reboot your system (without the disc/key in!) to Windows.
In the link above you'll find the latest Ubuntu build for desktop, and instructions on making a bootable disc or USB key if you don't already know how. If your computer is having trouble booting to Ubuntu, check the boot order in your BIOS settings. Once on a certain computer I had to make the default booting from the disc, which I was able to switch back once I was done.
Note: Some systems make it physically difficult to use Linux.
I don't know what's in your custom build, but a laptop I purchased (with Windows 8) freaks the heck out when I try to boot to Ubuntu after having installed it. Apparently this is a
hardware issue. Most computers shouldn't have a problem, but this is one way Microsoft can use to try to clamp down on its market share. At the moment it's rare AFAIK, but it does happen.