Yes, you can use leftover chicken. You can also clean the bones a little and roast them before using. You can also poach a chicken until the meat is just done (skimming at leasst twice); remove it from the pot; cool it enough to handle; clean the carcass; reserve the meat; roast the carcass; return the carcass to the pot; go through the skimming cycle again -- all before adding the aromatics. In the words of Sir Mick, "Time's on my side. Yes it is."
Roast-chicken stock, aka brun, is a wonderful thing. If you're going to keep one (there are three) type of chicken stock around that's the one to keep.
There are a few little details to making chicken stock, like how many times to skim; when to add the aromatics; when to remove them; and so on; but you lose the essence of the process if you think ingredients must be carefully measured, timers set, and every step followed just so. The most important thing is concept. Fortunately, they're rough, ready, pithy and easily assimilated.
Good stock is brewed. Just like tea it's an infusion, not a reduction. Watch your temperature, you want it very near but never at boiling through the entire stock making process. Keep it clean and clear. Don't hover over it stirring (and chanting). Once you have stock, and the carcass out, you can start thinking about jus, glace de viande, and various other reductions -- but not until then.
Stock is not soup. Rather, it's a base for soups and sauces. Remove and reserve any meat as soon as it's cooked. If you want chicken soup with chicken in it, you can return the meat to the stock later as part of the soup making process. Remove and discard the aromatics. If you want chicken soup with vegetables, add fresh vegetables later. No salt.
Hope this helps,
Edited by boar_d_laze - 3/31/10 at 7:44am