Not knowing the nature of the personal problems makes it more difficult to answer, but in general this is what I would do.
First, try to swap with another teaching assistant so that you are not in the same section as the person you have a problem with. Do not tell them the name of the student when you ask if they can swap with you.
Next discuss with the professor and see if he or she can arrange a switch or use someone else. This may result in less income for you if they have to hire someone else in your place. This may be the best choice though if the relationship is truly bad such as an ex-husband who abused you.
If the situation is such that the person is someone you intensely dislike but are not afraid of, consider behaving like a professional and leaving your personal feelings at the door. You could look at things like making the projects/tests/quizzes be submitted with only a number (someone else can keep the key until you have to submit grades), so that you don't know who submitted which ones until after all the projects are graded. You could ask someone else (like the professor) to review this particular person's grade to ensure you were being fair. That could help keep the grades fair.
You will also have to help this person if help is requested. Stay at a safe distance if need be, but answer the question exactly as you would handle it for other people. The perception of being fair is increased if you can say things like, "good question" when the person asks, so that you are not the one appearing negative even if the other person behaves in a hostile manner. A little praise can go surprisingly far when trying to defuse a personality conflict.
If the person dislikes you as well, then he or she might try to make you look bad by asking difficult questions. If you dig in and learn the subject far beyond the level that is covered in the class, then you can answer these questions easily and thus look knowledgeable when the initial plan was to make you look stupid.
If the difference is political or religious, don't bring up those subjects at all unless the course material requires it (hard not to discuss religion in a course on comparative religion). If the other person brings up those subjects, deflect by pointing out those are not appropriate topics for this class.