Let me answer the second question first as it is easier.... No, you don't need to have a post-graduate education in chemical engineering. BS graduates in chemical engineering are highly sought after by industry and other employers. This degree is essentially a "pre-professional" degree and you will graduate with all the tools you need to be a contributing member of the engineering community at work.
However, roughly 15% of a typical graduating BS class of chemical engineers will choose to undertake a post-graduate education. This is typically either in a professional degree program like the self-funded 1-year Master of Engineering degree which helps by either broadening your engineering skills or deepening them in one area of your choice. Some of the 15% will choose to apply for 5-year fully funded PhD degree programs so that they can dedicate themselves to mastering the ability to design and conduct independent research.
Now to your first question: "what average do you need to be successful in ChE?" I expect that by "average" you are thinking of a GPA. The truth is that almost all BS graduates in ChE will be successful in their post-graduation careers, regardless of GPA. Those with high GPAs may get job offers more quickly or get more job offers, but there is much more to success than just an undergraduate GPA.
Here are some of the other factors that lead to success post-graduation and few of them show any correlation with GPA:
(1) Your personality. Careers of any sort require that you have the type of collaborative personality that make you easy to work with. Be a "can do" person.
(2) Your ability to lead. Few are born leaders but you can seek out experiences at college that will sharpen these skills.
(3) Broadening experiences. Industrial internships can help you get that important first job post-graduation. But employers also value volunteer work and outreach activities that show that you care about community-building. Consider international experiences to demonstrate that you are aware that there is not just one culture in the world and that you can appreciate and thrive in other cultures.
(4) Communication skills. You can be the best test-taker in the world with razor-sharp math skills, but if you cannot communicate your ideas, experiences and vision to others then you will not be as successful as you could be. This is a critical skill to acquire. Most universities have excellent courses in written and oral communications.
(5) Mentoring and training abilities. No matter what job you do post-graduation, you will end up mentoring others or training others. Seek out experiences within and outside your major to learn how to help and train other people.
(6) Do what you love. You should be excited to get up every day and do your job. Keep this in mind at college. If you find that your major does not make you feel that way, then find a major that does. Your heart will tell you when you find the right one. And remember that dictum in life and you will be a happy and contented person.
I hope this helps. Keep up your math skills and sharpen your enthusiasm for understanding how the physical universe works and you will be a great engineer some day. Good luck