If you're looking to share, try spareroom.co.uk - create a profile for someone searching, reach out to people in your budget bracket and say hello. May help to do a month in AirBnB before so you can figure out what area you want to be in / meet the person/people you may end up living with.
If you're living by yourself, try rightmove.co.uk - you'll end up talking to a bunch of agents. Obviously you have rent / bills / council tax on top of what's on the site. Again, may be worth a month in AirBnB to try and figure out the area you like.
FWIW, I like East London - Shoreditch, Old St., etc. It's a pretty good blend of people and things, and a good location for doing stuff (or at least starting your journey). I'm in London - contact details are in my profile - feel free to reach out if I can help more.