IMO the league tables mostly tell you what kind of intake the school has, rather than how good the school is. The head at one of the Dulwich schools we visited admitted as much - they were top of some league table this year, but she said they won't be next year, purely because they had a lower ability intake. And the best predictor of academic performance is the social background of the kids.
There is no substitute for visiting the school yourself and seeing what you think of it. Word on the street can often be misleading as not everyone has the same criteria as you. Though I know it's hard if you don't know the area and are trying to narrow things down.
In the Denmark Hill area, the school I know of is Dog Kennel Hill, which has a good reputation. I live near Goose Green and looked at Heber, St Johns and St Clements and Goose Green. And would have been very happy with any of them. But I think I'm unusual in not particularly agonising over this decision! And in Dulwich Village I have heard good things of both Dulwich Hamlet and Dulwich Village schools.
I particularly liked Goose Green, which is a school that many middle class parents wouldn't even visit, because a few years ago it was in special measures and really struggling. It's now got a fabulous head and a lovely atmosphere.
In the end my dd will start at St Johns and St C in January, which was my dp's choice because it is our closest school. I didn't want a faith school, but was persuaded that I shouldn't let my politics get in the way of dd1 being able to walk to school without crossing a road .
Good luck. My experience is that the primaries round here are great.