Hey there HowlingDuck,
It sounds like you have signed in with, and authorized the computer with your Apple ID but you are still unable to see Radio in the Music library. I recommend checking to see if Parental Controls were set to block internet radio:
iTunes: Using Parental Controls
http://support.apple.com/kb/ht1904
- In iTunes, open iTunes preferences. Mac users: From the iTunes menu, choose Preferences. Windows users: On the Edit menu, click Preferences.
- Click the Parental tab.
- Select the checkbox next to any item you wish to disable or restrict.
Thank you for using Apple Support Communities.
Regards,
Sterling