Since the opening of the Docklands Light Railway extension to King George V, getting to City Airport without a car or taxi couldn't be simpler. There are frequent trains every 10 minutes or less throughout the day, and the journey takes 24 minutes or less depending where you start, and there are number of stations that link up with underground lines, and a big plus if you have luggage is that most changes on the DLR network are usually on the same platform, although because most of the system is elevated you will probably have a few stairs or a lift journey to get to the first platform.
Trains start in the city at Bank which is an interchange for the Central, Northern and if you like a bit of a walk the District line at Monument. Or alternatively there is Tower Gateway station (currently closed until Spring 2009) just around the corner from Tower Hill station on the District line.
The other stations to join from other lines are Stratford and Limehouse for National Rail services and Shadwell for the East London Line (although that is currently closed while they extend it), and also Canary Wharf for the Jubilee Line. In fact Canary Wharf is not a bad place to stop off, the buildings are very impressive and the shopping and places to eat are excellent. Gets a bit crowded weekday lunchtimes though.
The DLR is driverless, which means if you are lucky enough to get a seat at the front of the first carriage you can really enjoy the ride as much of the track rises and falls into each station like a giant concrete rollercoaster. Equally impressive is the curved tunnel out of Bank station, watching the lights on the side of the tunnel whizz by while the wheels squeal and the train rocks from side to side. Because of this there is usually a rush to get into the front seats once the doors open.
Even though there is no driver there is a 'Train Captain' on board to help, and check tickets so even though there are no ticket gates on most of the stations, buying a ticket is easy so there is really no excuse. Oyster readers are usually at the entrances and exits to the platforms but if you are like me you will struggle to remember to "touch out".
The station at City Airport is just outside the terminal building, and its only a short walk to the check-in desks.