Hi, I think the best way for you from Hangzhou Airport to Beijing is by air. For details information,including cost and time schedule, you can check China Flight Schedule(https://www.travelchinaguide.com/china-flights/) for recent information. You can visit Beijing and Shanghai in 5 days. The best way back to Shanghai from Beijing is bullet trains, just sleep one night, 10 hours on the train. 452 or 542 RMB for one train ticket. There is also one bullet train from Hangzhou to Beijing (south). You can go by this train instead of the airplane too. It is also a night train, arriving in Beijing in early morning. Can save some time.