Job Interview: Do I tell them why I want a new job?

I would just add a line in your cover letter, towards the end, like, "I am currently planning to relocate to CITY to live with my wife. No, say you're currently in the process of moving. Employers generally don't like to be your desperate foothold in a new city; they assume you'll use them to move without risk, wait till you're established, and then jump ship for a better opportunity. Act as if you're moving with or without the job.

Yeah this is like a built-in great reason to want to move. And "I was looking to relocate to ___ to be closer to my wife, which is when I heard about your company, which I am excited about for blah blah reasons" is not a bad segue.

It is true that it would give this new company increased leverage in salary negotiations; however, out-of-town candidates are often not taken as seriously as in-town candidates. This depends on location, industry, and company culture. This really depends on how risk-averse you are--would you rather relocate and potentially contend with a lower salary, or would you rather live apart from your wife while trying to get the highest offer possible?

It's only a salary problem if you let it be. Do your research and know what they should be offering, don't accept less than that. They may, on the first try, but just say no. My company is unbelievably shitty about hard-pressuring people to move across the country to live in a beautiful but expensive location where they have no family or non-work reason to be, often in three weeks or less, and are SO SURPRISED when they're gone in less than three months. Whereas my one hire, who came from the east coast but his parents live in town and he grew up here and went to college an hour away? Still here two years later. Tell them your wife already works in CITY, absolutely.

Family reasons are a perfectly normal (and good) reason to be looking. It's perfectly reasonable to tell them that's why you are looking. Of course make it clear that your interest in them is about more than just location.

Why would you think it's weird? Do you mean that there's something unconventional about living apart from your wife to begin with? You don't need to get into detail, just say you're moving to City to live with your wife. (As a company recruiter, I'm always happy to hear that a candidate from out of town has personal reasons for wanting to be in my city, rather than that he's just applying to us on a whim, and may not be serious about moving.)

Personally, in future I would use your wife's address on your cover letter so they can see you are committed to living in the area. For this interview, yes, I would say you are moving to [wiife's address]/new house in wife's city.

One of the great things about your situation is you are absolutely free from having to say anything negative about your current employer. You can always toss in a "running out of places to promote up to", which is a pretty good complaint to have about your current employer, but needing to move means you're free to talk about how much you love your job and are so sorry to be leaving but it's time to move on and you're so excited about living in CITY and just being the most positive applicant they talk to. Take advantage.

Just wanted to echo that your reason is perfectly valid (in fact I recently moved cross-country because my fiance got a job here). Also rethink your assumption "I know they can't ask about my personal life and reasons for relocating," because that is just untrue. They can ask you pretty much anything and decide to hire or not hire you for pretty much any reason, unless it's some kind of discrimination (based on your race, gender, etc.).