It'll depend very much on where you want to be. I've put a link below to places around East Finchley tube station. This is right on the Northern Line, straight into central London, takes about 25 minutes. You're looking at about �150 per week minimum round here, and Council Tax will be about �600 plus a year on top of that.
http://www.rightmove.co.uk/action/publicsite.P ropertySearch;jsessionid=3C068B5EE437F81B9454A 4667B02C256
I've had a quick look at other areas of London and it looks like you'd struggle to find any studio or 1-bed flat for less than �100 per week.
I hope this helps you
Good luck