Use time machine for backing up a different computer
To exclude your internal drive from backup, do the following. Go to System Preferences > Time Machine. Click Options... Under the section titled "Exclude these items from backups", click the + button. Select your internal hard drive. This will prevent Time Machine from backing up your internal hard drive.

If your external hard drive is working just like you'd like your internal hard drive to work after you finish fixing it, you can also restore your external volume onto your internal volume. To do this, boot into Recovery Mode (holding R upon boot). Once booted into Recovery Mode, follow the steps in this Apple Support article, under the section Restoring the backup disk image's contents to your internal Mac OS X disk. You can begin with Step 4, as you have a newer operating system than the article was written for.