Also click on the By Train Or By Bus link at the bottom of the page. Not sure about airports - Southend's the nearest by the looks of it but it's on the other side of the Thames estuary. Gatwick's 50 miles away. There might be smaller airstrips accessible by light aircraft if that's what you want.
Chatham is just outside of the M25, so any of London's airports are reasonably close. (London City Airport is the closest but it's mainly used by full-fare airlines, rather than budget ones. Travel via Gatwick would be a good route, as you can get directly from there to Victoria Station and then catch a train from there to Chatham).
Further thoughts:
Chatham's Historic Dockyard is a big site and takes quite some time to get around. However I'd still rate it as a 'long afternoon' visit, rather than "a couple of days".
If you're thinking of staying around there, please be warned that much of that part of Kent can be VERY depressing! However Rochester is quite a pleasant town (with lots of history) and it's definitely where I'd choose to stay around there.
There is little worth going to Chatham dockyard for... I include the Dickens experience. a really sad little shopping mall and the usual eating places...and a tiny cinema.... Rochester is much much more interesting and well worth a visit.