How can I make a method private in an interface?

An interface can only have public methods. You might consider using an abstract base class with a protected abstract method AddError for this. The base class can then implement the IValidationCRUD interface, but only after you have removed the private method.

like this:

public interface IValidationCRUD
{
    ICRUDValidation IsValid(object obj);
}

public abstract class ValidationCRUDBase: IValidationCRUD {
    public abstract ICRUDValidation IsValid(object obj);
    protected abstract void AddError(ICRUDError error);
}