Where can I stay in a castle in northern Wales, specifically in the Snowdonia Natioal Park, Anglesey or Caernafon-Llandudno areas (not sure of the official name of this area)?
There are actually several, as found in a quick google search. But one that matches your area certainly seems to be Bodelwyddan Castle. It is just under an hour from Chester, 10 minutes from the seaside town of Rhyl, and half an hour away from the Victorian Pier at Llandudno.
A similarly close one is Ruthin Castle Hotel, although it's a bit closer to the Chester side of North Wales.
I believe the area names you're looking for might be Denbighshire, which both these castles are located in.
In addition to Mark's suggestions, there's Bryn Bras Castle, 10 minutes drive from Snowdonia National Park and Caernafon.
And also Glandyfi Castle on the southern edge of Snowdownia National Park.