I can chime in on rental prices since I just found a place in Clapham (just north of the Common, so I'll have a lovely walk through the Common to the tube station every day). One bed flats are nearly impossible to find at the low end of the price range, but it can be done with some persistence. I lucked out and liked the first place I saw in Clapham, despite it being unfurnished. I'm paying £225 per week, which is £975 per month. (General rule of thumb is 4.3 times per week rent gives per month rent.) Shelled out about £1,000 at IKEA for furnishings (including home delivery) and will be going to get a television today for under £300. I'm here for two years, so the money I'm saving by having negotiated the landlady down since the place was unfurnished will cover the cost of buying furniture. Finding a place for £1,000 per month in Clapham is possible, but if you can spend more than that, you should have it pretty easy. Just be sure to find out what the monthly rail cost is to your job so you know how much you'll be paying for that. Probably less than having a car, though! (I work in Central London and my fellowship sponsor pays for my tube pass, which helps a lot.)
In London right now, you cannot believe the property listings. Everything is gone before it even makes it online. If you're interested in Clapham, use Rightmove and FindaProperty to find estate agents covering Clapham (I used Keating Estates but also talked to Bushells and Black Katz) and start calling them every day to see what they have. Know what you're looking for and how much you can spend (give them the weekly rate), and they'll tell you if they have anything. If you see something and like it, you need to put in the offer right away. I was going to see some places last Monday, but by the time I got to the estate agent, two had already been put under contract (from morning to afternoon).