When my nokia lumia 1020 is connected to my PC is it running off my internet or using my data?
That depends on how you mean "connected" -- if you have it connected via USB, then no. It is still using mobile data. If you are using software like Virtual Wifi Router and your phone is connected to that SSID then WiFi will trump mobile data and it will go out that way.
This used to be a built-in feature of Windows Phone 7, but as outlined by another answer, its gone from WP8+.