Brendan,
From London City airport the DLR (light rail, but basically all part of the integrated London transport system) interescts the tube at Canning Town (Jubilee line) or Bank (loads of lines).
If you are going to Central London I would suggest changing at Canning Town and getting the Jubilee line over towards Westminster, Green Park or Bond Street.
You say you will be getting the Picadilly line - where is your actual destination ? If it is in West London (say west of Earls Court) then it would make more sense to fly to Heathrow and just get the Picadilly line from there. Otherwise you would fly to London City, get the DLR, get the Jubilee Line to Green Park, and then get the Picadilly line further west. If you are on a day return, this transfer in both directions will take a fair amount of your total time.
If you are going somewhere on the Northern part of the line they you would be better advised to get the DLR to Bank, Central to Holborn and then Picadilly to wherever you are going.
It all depends on where exactly in London you are going.
z
p.s. you probably have this, but here's the full map that you need for connectivity -
http://www.tfl.gov.uk/assets/downloads/standard-tube-map.pdf
p.p.s. to answer your actual question - does the DLR just hook up with the tube - yes, it's all integrated so you just buy one ticket (well, a day pass works but I'm sure individual tickets are the same), get on the DLR, get off, change platform, get on the tube, leave station . . .