The cheapest way is cetainly not the easiest way. If you will be there first time and do not familiar with it, suggest you choose easier ones. Guangzhou East Raiway Station has trains to Hung Homg (about 190 HKD), then change to Bus A21 to the airport, 33HKD.

Get to Shenzhen by train from Guangzhou Railway Station, take subway (about 33HKD) from Lo Wu to Hong Hom, then change to A21. Seems the second way is cheaper.
Hope it helps.

Thanks a lot Ms Emma. I really appreciate your help.
I've heard that there is also a bus from Guangzhou that goes directly to Hong Kong Airport. How about this one? Can you please tell me how long does it take to get there (HK airport) and where is the bus station ?

Find the bus to HK Airport at Guangzhou Garden Hotel and China Hotel. It take about 2.5 hour and the ticket fare is about 210rmb. Hope it is helpful. :-)

Thank you very much Emma!