For SHA-PEK flights, you can look at www.ctrip.com
as they always have good deals for China domestic flights.
PEK-HKG I might suggest any Chinese airlines since they are probably cheaper than others like CX and KA.
HKG-SIN Jetstar Asia or Tiger Airways. Might want to try flying from SZX or MFM. Last time I checked Tiger's fares from SIN-SZX-SIN was only roughly $230 SGD return, including all taxes!
SIN-BKK lots of carriers from Thai AirAsia, Thai Airways International, Tiger Airways and Jetstar Asia. Thai Airways might be a tad expensive so you might want to consider the other 3. CX and SQ is definitely out of the question if you are looking for cheap deals. However, SQ sometimes give very attractive offers for SIN-BKK-SIN for around $388 SGD.