Oh now this is my question :)
If you like parks, there are the 'tourist' ones in town - Hyde, Regents, St James and so on. But take the tube down to Richmond and walk up the hill to Richmond Park. Firstly you'll get a great walk along the Thames on the way up, and then you're in the biggest open air space in London. Originally a hunting ground for King Henry the 8th, I believe, he built a wall and kicked the peasants out. Fortunately today anyone can go in, and there are still 2 herds of deer - 300ish fallow and 300ish red, plus squirrels, foxes and more. It's beautiful, and the view from Richmond Park to St Paul's is so special that it's protected by law!
For clubs, um....Clapham Common has a great number of bars and so on around it which won't get the tourist crowd, but you may get a lot of Kiwis and Aussies who live in the area.
For a market, try Borough Market - it's near the London Bridge tube station, Borough High Street exit, open from Thursday to Saturdays in the mornings. Lots of great food, a little expensive perhaps, but you can actually do well by eating the free samples ;)
One way we found to really get 'local' Londoners was to catch the long tubes to the end of the line - Cockfosters, for example, has a fantastic pub called the Cock and Dragon, and you're practically guaranteed not to find tourists there! Epping is similar, but you may experience the music-stopping-and-locals-staring feeling as you walk through the door of the pub.
London is huge, and so cosmopolitan that it's hard to even define what is a Londoner these days! But it's a great city and there's lots to find and explore.