The Chinese Academy of Social Sciences (CASS) issues annual reports on China (the Blue Book) and China's standing in the world (the Yellow Book). The Yellow Book measures the economic, social, diplomatic and military power of eleven major nations (Brazil, Canada, China, France, Germany, India, Italy, Japan, Russia, United Kingdom, United States), along with the potential and current trajectory for each nation. CASS rates the United States as the most powerful military power, followed by China (with one third the power) and Russia. The overall ratings put the U.S. in first place, and China in seventh.
Many Chinese believe that CASS rates Chinese military power too high, which is done by concentrating on just counting weapons and personnel, and leaving out the quality of training and leadership, and completely ignoring corruption. This is also a problem in the Blue Book, but that's politics for you. China is a communist police state, and all media is normally obliged to make the state look good. Corruption is not ignored in these reports, but its impact is played down.
While the Yellow Book draws mostly on open (publicly available) sources, it also uses internal government material (analysis of foreign nations). The Yellow Book is meant to inform a large number of educated and influential Chinese what the government thinks of the major world powers.
China is respectful of American economic and military power, without seeming to understand where it all comes from. For that, and other, reasons, China is wary of the United States. The official line is that America is in decline, and China is growing. In a few decades, China believes it will be more the equal of the United States. On the street, the outlook is different. Corruption is a major problem, as is pollution (which is sustained by corruption) and a growing worker shortage (brought about by a "one child" policy that was introduced 32 years ago, which prevented about 250 million births). That, plus the availability of cheap determination of the gender of unborn children (which caused over 20 million females to be aborted), has caused a labor shortage and a growing age imbalance (more older, than younger, people).
The recent global recession, Chinese economists noted, put seven million Americans out of work, but cost more than four times as many Chinese to lose their jobs. China's economy continued to grow, but many Chinese (and foreign economists) don't trust the government statistics. So while China may be catching up, it cannot ignore the many uniquely Chinese problems that could bring that economic growth to a halt, or worse.
Meanwhile, it's American, not Chinese, culture that still dominates the world. The Americans invented the Internet, and although Chinese are the largest group of users, the net is still very much an American place. This "hegemony" bothers China a great deal because America also has lots of friends as a result. America allows people from all over the planet to come and become Americans. Most other nations don't even try this sort of thing, and no one does it as well as the Americans. Many Chinese have availed themselves of the opportunity to become Americans. The few who have returned home to make their fortune, largely speak fondly of their lives in America. Few people do that with China. Even the "overseas Chinese" communities throughout the region are wary of the current government in China, while still embracing Chinese culture. So do the math. China has few friends and allies, America has many. China would like to play the traditional diplomatic strategy, by assembling a coalition of like-minded allies to oppose the big guy (America). That doesn't work, as too many nations are more inclined to distrust China, than America.
China is also uneasy that America is so different as a superpower. American might is based more on economic, financial and cultural, than military power. Although the United States has the largest military in the world, those troops are often involved in disaster relief, or removing tyrants (like Saddam and the Taliban) from power. When the American military rolls in, they eventually leave the "conquered" nation better off. That sort of thing does not compute using the ancient Chinese foreign affairs calculus.
The Yellow Book leaves you with the impression that China sees America as more powerful than most Americans do. That is largely true, and accounts for even more inscrutable (to Americans) Chinese behavior when the two nations get into disputes, or just try and negotiate solutions to problems the two of them have.