It kind of depends on who the person is. If it's a close relation such as a parent or sibling then you need to decide if you want to be part of the family or if you want to be an outcast. If it's anyone else just make it a point to be busy when there's a gathering with that person in attendance. If you go to a gathering and the person is there when you thought they wouldn't be, leave.
There are lots of ways to leave a gathering without hurting your host's feelings. Spill something nasty on your clothes and excuse yourself to go home and get cleaned up. Have a friend call and say they have an emergency and need you to pick them up. Take your host aside and explain you've been having a bad bout of diarrhea that you thought was over but it isn't and you need to leave. These are just a few and they all work. One reason they work is because if this is a family gathering then whoever invited you without telling you about the other person being there knows you don't want to be there and, unless they're not any nicer than the person you're trying to avoid, will let you leave gracefully.
Those are some ways out for you but there's another alternative. Turn the tables on the person you don't like. If they're always looking for flaws, give them something to talk about and when they comment, turn it into a joke by laughing at yourself about it. When commenting back to them start your sentence with "I know you hate me but ...". Say it very nicely and with a smile but put it into words. That will let the people listening know that an animosity exists and that you're making the best of it while the other person is poking it with a stick.
Lastly, if this is something that is condoned by your family then you really need to consider the outcast possibility. It usually isn't forever but if the family doesn't see that there is a problem then avoiding all of them may be the only way to get them to acknowledge the fact that there is one.
Some people think that because they're family they have a right to say anything they want to you and you have to forgive them for being rude, unfeeling, and downright nasty. That's not true. People who treat you or anyone else like that usually are sadly lacking in self-respect and without self-respect you can't respect anyone else. They are sad people and you should pity them--and probably tell them so.
Yes, I have a mean streak...