Do you need to know how to calculate the equation of a circle given 3 co-ordinates?

Yes

I was looking over C1 on exam solutions and there's a video about calculating the equation of a circle given three co-ordinates:

http://www.examsolutions.net/maths-revision/core-maths/coordinate-geometry/circle/3-points/tutorial-1.php

I was wondering if this was actually on the AQA C1 specification?

All of the knowledge required to answer that question is covered in C1 so whilst that question isn't explicitly mentioned in the C1 specification, it could come up in an exam since C1 equips you with all the information required to answer it.