For most freelance work you do not need a formal resume. You may want to include a list of clients if you are an established illustrator. If you are just beginning in your career then don't worry about a list of clients - let your work speak for itself.
Do I need a college degree?
This depends on your learning style. Some illustrators have degrees in art and have found art school to be a valuable resource and learning situation. For me, that was the best choice and I was able to take classes in a wide variety of areas including painting, drawing, life drawing, sculpture, woodworking, metal work, jewelry, weaving and ceramics along with many classes in illustration and a regular university education. If you do choose to go to school for illustration look for a school that has a strong program in both basic art skills and illustration classes.
Other illustrators are self-taught because they learned best that way. Other illustrators have taken a middle road by taking classes but not finishing a degree. You need to decide what works best for you. Most illustrators end up learning new skills during their whole career. Very few schools offer courses in the business of illustration so you will have to teach yourself those skills on your own. I think the one of the most valuable classes for illustrators is life drawing.
Whichever road you take you should also have a back-up career in mind. Very few illustrators can start out working full-time in any illustration field unless they happen upon a staff position, which are few and far between. You are going to need some other career to support you while you develop your career as an illustrator. If you are young, then concerns like dental insurance and health insurance may seem inconsequential but they really are important! Some freelance illustrators have their second or part-time job in a creative field but other illustrators find that working in a non-art related field helps keep their creative level high for their illustration work.