I grew up fairly close to Belvedere, in one of the nicer parts of the South East London suburbs. My parents had grown up in Bexleyheath and we had family all over the immediate area. I spent my entire teenage years desperate to get out and I swore I would never, ever, no matter where my life circumstances took me, live there as an adult.
There is literally nothing going on, ever, apart from the occasional excitement of some gang violence that spills over from the more 'dodgy' surrounding areas. The South East London suburbs are populated by people who spend their weekends at B&Q, eat in Pizza Express, drink in Wetherspoons, and who think Costa coffee is the height of sophistication. The streets are filled with a depressing straggle of identikit, over extended 1930s houses and the high streets consist of nothing but bland chain restaurants, hairdressers, and estate agents. There is no culture and no real community. For me, the South East London suburbs are what hell looks like, and living there is like being slowly suffocated by the hands of mediocrity. You couldn't pay me to live there.
If you need to commute to London and can't afford to actually live in London, then go much further out and live in the Kent countryside proper. As the trains are fast, you can often get into London in the same time or even more quickly as you would from a suburb. And you'd have a much nicer view, an actual community, and something to do on the weekend.
I hate to be harsh, and I know I'm a total snob, but I spent eighteen years of my life dying of boredom in South East London suburbia and I wouldn't wish it on my worst enemy. For your budget, go out to some of the lovely villages surrounding Sevenoaks or Tonbridge - your money will get you a beautiful cottage, access to stunning countryside, and village community with a lot going on, plus a 30 minute commute on the train into central London. The best of both worlds!