I already know that there is Sir Robert Hotung. But I am speaking of immigrants.
This is a very broad question, since there was a marked difference between the way Chinese people were treated in Victorian times across different parts of the world, not to mention different decades. In Canada, for example, the history of Chinese migration is most famously aligned with the history of the railroad, but Chinese people also worked in mining (they stayed in the gold rush long after others left), farming and canning. In other areas, Chinese men carved out a social niche by doing jobs traditionally considered "women's work" (like laundry services) in areas where there were few women to do the job. By playing on the stereotype of Chinese people being more 'feminine' than white men and providing a service that white men did not want to do, they were able to make a living, often a good one. Other Chinese immigrants rose to prominence by exploiting their fellow countrymen and acting as brokers/traffickers to lure them to Canada to work on the railroad, or dealing in opium, or running gambling dens.
In places like Vancouver, the relationship between Chinese immigrants and Western people was fraught and shifting. Incidents like the 1907 Anti-Oriental Riots highlight how complicated race relations were in this time. Still, several Chinese businessmen rose to prominence and founded the Chinese Consolidated Benevolent Association, which represented Chinese immigrants in legal matters, sent their bodies back to China if they died, etc.
While I'll leave it to others to detail specific examples of Chinese immigrants who rose in social standing among white people, many immigrants focused on building their social standing within Chinatown society, which was a thriving society in its own right. Many felt that it was simply too dangerous to try to gain standing in the very racist upper-class Victorian society of the time.
In short, it was possible for a Chinese immigrant to gain social and economic capital, but it was difficult because the minute too many immigrants reached the middle or upper class, racist laws like head taxes or flat out riots would try to strike them down.
I can give an answer for Australia.
The short answer is that it was possible if not likely to occur. In order to be accepted you basically had to assimilate as quickly as possible. The most well known example in Australia was a man known as Quong Tart who in the late 1800s s was a highlight of the social scene in Sydney. He was a tea house owner who always jokingly said that he really should have been Scottish and always wanted to be called McTart. He became an accomplished cricketer married a school teacher and many children who became the first Asians to attend the prestigious PLC in Sydney.
To quote some of the obits that accompanied his death.
"He was whiter than hundreds of the class folk who were wont to sneer at him because he was a Chinese"
"Putting our well-advertised politicians on one side, probably the best known Australian at the time of his death was Mr. Quong Tart."
http://trove.nla.gov.au/ndp/del/article/126441583?searchTerm=quong%20tart&searchLimits=
http://trove.nla.gov.au/ndp/del/article/111949545?searchTerm=quong%20tart&searchLimits=