First, consider it may be true. At least, your child believes it is true. That is a good place to start. Do not start by telling her it is not true. She will only see you are unwilling to see her side of things. You are just like those teachers who are against her.
Children who are doing well in school almost never have this complaint. Even if they sometimes do, it does not keep them from doing their work. It may be human nature to feel a little put upon at times, especially when your child has been criticized or did not get something she wanted.
Think about this. An adult gets reprimanded at work or does not get a job he thought he deserved. Do you think you would hear him say this? "The people at work are always reasonable and fair. I had the reprimand coming. I appreciate their honesty with me. I am a better person for it." Children usually handle these kinds of things better than adults but even they have their limits.
Say, "Okay, the teachers have it in for you. You have told me they are not doing anything specific; but you can tell. Maybe they just don't like children or at least some children. Have I got it right so far?" You then continue, "They have it in for you; and it keeps you from doing well, from getting the education you tell me you want. Here is the part I do not understand. As angry as you are, it surprises me you are going to let them win, let them stop you from getting an education. It looks like you would fight back. Do you know how to fight back, how to win?"
Listen to see if his ideas would help or make things worse. Likely he will not have an idea. You then say, "I have some ideas but I think you would rather be angry and lose. I wonder if you have what it takes to win. What do you think?" If he chooses to quit, it was worth a try. His problem goes far beyond feeling picked on.