I am updating my resume. I am currently designing a website for my client’s client. My client has a horrible website. If I put my client’s name on my resume I worry that the resume reader will thing I designed their horrible website. But I didn’t. I am designing their client’s website. Whose name should I put down on my resume, my client or my client’s client?
Of course it makes sense to list only the site you actually produced. You want to provide the details of the work you undertook in the portfolio and there you can state (if you feel its necessary) the client where you got the project. It’s not your fault if your client has an awful website but you’ll only need to hold responsibility to whatever work you produce. So just carry on as you are and skip over that ugly website which you had nothing to-do with! 
In this case it would of course be preferable to put the client’s client website on your resume - for the obvious reasons.
You need to clear with your client however if it’s okay if you do that. They may be under the assumption that you’re white labeling for them, and therefore might not like it if you put it in your resume.
Yes, you should mention client’s client address, because employer should know about your work. Even you can write in Resume as a note how you explained in this thread.
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Put your work/s(client’s client) but mention whom you are working too(client).