I expect that forking a reddit client is the way to go for UI (if you don't have any in mind, I think AlienBlue and Reddit is Fun are probably worth looking into for this).
For the backend, reddit exposes itself through json, which LW doesn't seem to; e.g. http://www.reddit.com/user/philh/.json works, but http://lesswrong.com/user/philh/.json (and http://lesswrong.com/user/philh/overview/.json ) don't. I expect clients to mostly use this, so you'll need to rewrite those portions of the code.
Turns out AlienBlue did release their original version as open source, but the code is four years out of date! Hmmm.
Yeah, I would probably end up scraping the HTML. I filed a bug about .json being broken two years ago, but even if it were fixed, it seems that LW has quite a few customizations that the JSON output likely has not caught up to...
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one.
3. Open Threads should be posted in Discussion, and not Main.
4. Open Threads should start on Monday, and end on Sunday.