To get from the Eurostar to London City Airport, you can take a combination of the London Underground (tube) and the Docklands Light Railway (DLR).
From the Eurostar, you will first need to transfer to the London Underground in St Pancras Station. Take the Northern line from St Pancras to Bank or the Circle line from St Pancras to Tower Hill.
The trip will take 9 minutes on the Northern line and 15 minutes on the Circle line. Tickets cost £2 if you get an Oyster Card at St Pancras station.
From Bank or Tower Hill, you can connect to the DLR to get to London City Airport. From Bank, you can transfer inside the Underground station, whereas from Tower Hill you must exit the station and follow signs to the Tower Gateway DLR station.
Once on the DLR, take the train to the London City Airport stop. Make sure to board the correct train, as there are multiple branches of the DLR. You can always change before the branches split if you board the wrong train.
The journey will take 25 minutes from Bank, and the service is direct. It will take 28 minutes from Tower Gateway and requires a change at the Canning Town DLR station en route.
You can use your Oyster Card for the DLR. The price of the journey will depend on whether you travel at peak or off peak times, but should cost no more than a few pounds either way.