I started expats.askbot.com because I was getting tired of waiting for the Immigration Expatriates proposal on StackExchange to gain traction.
Since I was already a member of several internationalization and expat communities, I knew I had a large pool of "seed" users.
Initially, I contacted about a dozen people directly to start using the site right away. I got some great feedback, and they made a few posts.
Unfortunately, I waited too long to open the doors to the public because I was waiting on the beta invite codes feature. As a result, my site lost a lot of its initial momentum.
I still put links to my site wherever I can. If I'm on a discussion board, and somebody's asking a question that already has an answer on my site, I'll post a link. I've got a link to my site on all my online profiles. Etc.
My next steps:
I'm tentatively exploring promoting my site on some of the StackExchange communities (in my case, Travel.SE is a really good candidate). I came across this meta post the other day that seems to indicate that the SE team doesn't have a problem with clones (or at least that there's nothing they can do about them :P).
Google's AdWords team sent me a $100 promo credit and a free consultation (I hear they do that sometimes if you have an Analytics account). I'm not sure if I'm at a point where I'm ready to start spending real money on this project yet, but I might try it out for a month or two and see what happens.
I've been thinking about hiring a VA agency such as 123employee.com* to do research and post answers on my site. In addition to getting more content on the site, I'm hoping it will also help make the community look more active and get more people interested in posting.
* I only list this vendor here because a friend of mine already uses and recommends them; I've never used them myself before, however.