"when she gave me the bill you'd of thought she had my appen

you'd of thought = you'd have thought = you would have thought.后者弱读时跟前者读音一样.作者为了表示说话的人是这样说的,就把句子写成了现在的样子.如果写成you would have thought或者you'd have thought读起来就不一样了,也就跟实际说话的效果不一样了.
再问： 谢谢答复，这种用法在美国文学作品中常见吗？
再答： 比较常见。类似的有模仿黑人的I ain't have no idea之类的句子，还有McDonald's的广告语I'm lovin' it中的lovin'也是类似用法。