During that period, weather in Beijing and Shanghai will vary from warm to hot. The average temperature will vary from around 22C to 26C or a little higher. In spite of that, there is somewhat temperature difference in the daytime and at nights. So a light coat and long-sleeved shirt are both needed. You need also prepare summer clothes.
For Tibet, big temperature difference is the distinct feature. The temperature in the daytime may be around 25-29C, while at night, it falls to only 0-5C. Therefore, apart from some necessary summer clothes, you also need take along an overcoat to keep warm. In general, the weather there will be a little dry during May, but become hot and moist in June.
Hong Kong will be hot then. The aveage temp varies from 25C to 28C. So summer clothes are terribly needed. From late May till June, it will be extremely hot and humid, with thunderstorms and occasional typhoons. Thus I suggest you pay attention to weather report before outing.