You'd need to have a PGCE in the area in which you want to teach. Different universities offer areas of study, but I'd guess you'd be looking at an Art teaching one. Again, how likely you are to get in on a Graphic Design degree would depend on the individual university.
For any teacher training course that qualifies you to teach in state schools, you would need to have GCSEs in Maths, English and Science, plus a few others (or equivalent), and preferably a few A levels as well.
The criteria are less strict for a PGCE for teaching in further education, and the course I did (at DMU) allowed me to pick my own specialism. I'm not sure whether it's still the case there, though, and I wouldn't know about the others.