var that = this; Keeping a class in scope of nested functions
Your code is fine (assuming your nested functions are merely proxies to the class methods), I would suggest if the only thing you are doing in the anonymous functions is calling the callbacks then you should just pass them in:
$.post(this.getRequest(), this.getUrl()).success(callback_one).error(callback_two);

There is nothing wrong with the use of that in this code.
Edit (after reading Esailija's answer)
You could also do the proxies mentioned in the Esailija's answer inline like so:
myClass.prototype.myMethod = function () {
    $.post(this.getRequest(), this.getUrl())
        .success($.proxy(this.myOtherMethod, this))
        .error($.proxy(this.myErrorMethod, this));
}

Essentially that is what you were doing with the nested functions; this just tightens the code by moving the proxy creation into a utility method.