How can I persaude my 325 pound cousin to lose weight with me without hurting her feelings?

Difficult to say. First of all, how well do you know your cousin? Secondly, how do you think your cousin feels about her weight? If she doesn't seem to mind being morbidly obese than it may be very difficult to persuade her. If you seem to think she is unhappy, or even depressed about her looks than it may be easier than you think to persuade her to begin losing weight. However you must tread lightly. Continued.
Part 3 of 3 (sorry) This is the last little part of my answer to your question (see previous 2 answers for parts 1&2). If telling her that you are concerned about her for all those reasons makes her seem interested, ask her to make a list of 12 reasons she would like to lose weight. Finally, with the amount of weight she needs she would really benefit getting on some kind of medically supervised program. Good luck!
Finishing answer. See previous answer for initial part of my answer to this question. In order to lose weight, the person must have the desire to do so, not because you or anybody else wants her to do it and she must be commited to do so for her own reasons. Let her know that obesity causes an increased risk of heart disease, diabetes, stroke, high cholesterol, sleep apnea, arthritis and many other diseases.