I would suggest that you have a look at three sites, Bing Maps, to view and study Ordance Survey maps. Then
www.nationaltrail.co.uk, The North Downs Way is there plus it offers a Google Street view of the route, walk it by sitting in you armchair at home. Also
www.gps-routes.co.uk is very good, loads of routes, county by county, plus Ordance Survey maps as well. If you want to hike in the South of England, look up the walks in Kent, Surrey, East Sussex, West Sussex and possibly Hampshire.
Lots to choose from but a couple of trails springs to mind which makes good use of trains. London to Gatwick Airport, start to walk the Sussex Border Path. London to Horsham, start to walk the High Weald Trail. London to Rochester, start to walk the North Downs Way. I'm assuming you will be making your way back to Dover?
For a circular walk, London to Tunbridge Wells, stay there and walk the Tunbridge Wells Circular Walk, use the train again and start at Groomebridge or Tonbridge.
August will be a good time for any of them, lots of apple, pear and plum orchards to walk though and to scrump as well!