What is the best way to do Leeds Castle, Caterbury and Dover?

We had a rental car - not sure if you will. We did all three of these locations on separate days on trips there - no organized tour - did on our own and really enjoyed each place. (except, we did take a tour at Dover through the WWII tunnels - that was worth it - booked when we were there)

We had a rental car and stayed in Tunbridge Wells, from where we easily toured the places you noted, combining Dover and Canterbury in one (longish) day, though in Dover we just walked along the cliffs until the wind and paranoia both picked up. We had done our research, had the GPS and it was quite easy.

Leeds, Canterbury and Dover is a very full day and tours may not provide what you are expecting.

Leeds Castle is priced and is run as a kind of theme park where you spend most of the day with a constant stream of 'events' theme park style.
The actual castle only takes about 20-30 minutes to walk around, not a lot for your money if you're only there an hour or less.
The day tours from London have a special access arrangement whereby the tours enter Leeds Castle before opening time and I presume the tour operators get a much cheaper price.

Dover can also be a dissapointment. What most people think of about Dover is the White Cliffs or Dover Castle. The coach tours don't visit Dover Castle because it would occupy too much time. You can't really see the White Cliffs if you're standing on top of them.
The coach tours visit to Dover is normally either a short walk aling the top of the ciffs or a stop at a pub which can seem a cynical device to sqeeze more cash out of the passengers rather than provide a worthwhile experience.

Canterbury on the tours is really code for a shuffle around Canterbury Cathedral.

If you're time poor and really must see all 3 then a day tour from London is probably the best option.

Otherwise I'd split the trip into 2 days.
There are several trains an hour on the London - Canterbury - Dover line. Best do your own thing and research local options in each town.

For Leeds Castle, there is a daily excursion bus by national Express to Leeds Castle that spends a good few hours there before returning to London.
Otherwise hire a car and explore the countryside around Leeds Csatle. Places like Hever Castle, Chartwell, Sissinghurst, Bodiam Castle are in my opinion more interseting than the supposed 'A' list tourist places of Canterbury and Leeds Castle.