There was two main phases to the racial relations in Hong Kong.

The first was prior to the Second World War. There was legal discrimination against the Hong Kong Chinese, in that there was a colour bar against them going to certain areas, and gaining more than the bottom ranks of the governmental service. At this time however, very few Hong Kong Chinese or Britishers thought this was a problem as the communities tended to remain strictly self-segregated, and outside of a (comparitively) small 'indigenous' community, both the Chinese and British tended to see living in HK as a temporary thing, and it was much better to focus on making a fortune and returning home than getting involved in politics.

The main practical limit on the Chinese was the colour-bar for entry to Peak and the island of Cheung Chau, even so several of the wealthiest Chinese were allowed to reside there, as the distinctions in society were much based much more on class (with the Britishers making up the upper and upper-middle, the Chinese making up the lower-middle and working classes). A sufficiently wealthy Chinese, with a good education and deportment could effectively 'transcend' the legal limits of their race in Colonial society.

Few of the organisations you would expect tried to speak out against this colour-bar, such as the Communists, as the 'benign neglect' by the British of the Chinese community made Hong Kong a haven for political refugees from the mainland, and many intellectuals, when repressed, remained in HK for a time. Some educational facilities were set up for the HK Chinese in the 1910s, with UHK being founded in 1911, and a government-sponsored schooling programme setting up 279 schools by the 1930s. This meant that some educated Chinese began entering the technical jobs of government administration, though numbers remained low.

I'm not trying to paint an overly rosy picture of the place, Victorian attitudes remained largely in place until WW2 and there was no kind of safety net for the Chinese, who were overseen by only a handful of officials and had no political representation, but as said before, few enough of the Chinese community thought of the place as home and focused on business and their own community.

After WW2, things began to change. Firstly because the 'superiority myth' of the white man over Asian peoples had been shattered by Japanese defeats of the Colonial powers. Hong Kong had seen a puppet Chinese regime governing the territory during the Occupation, some of the members being part of the former colonial administration, and some returning to it after the war's end (they had argued that it was a necessity that some qualified and experienced people remain for the benefit of the million or so inhabitants). The occupation was harsh, and expropriation was systematic, resulting in a halving of the population.

However during the war, the HK Chinese had played key roles in the BAAG (British Army Aid Group), who helped prisoners escape the Japanese, coordinated resistance, and provided aid and support to those left behind. One of the most prominent Chinese BAAG members was appointed as an 'unofficial' to the Executive Committee after the war, to help with rebuilding the colony, and official apartheid policies were revoked.

With the victory of the Communists and the onset of the Korean War, travel between the colony and the mainland became harder and a permanent Chinese population began to emerge. This meant that those who were disaffected with the government tended to seek change, rather than move elsewhere. There was also the problem for the British from this time that if they were not responsive to the needs of the HK Chinese they risked the ire of the Communist regime and the takeover of the colony, which militarily they were no longer in a position to resist.

Through the 1960s and 1970s the economy boomed. Many Hong Kongers became part of the property owning middle classes. Though many infrastructural projects and social policies were introduced in this period, a substantial minority were unhappy with their lack of representation which culminated in the Star Ferry Riots of 1966 and the more widespread rioting of 1967. From this time the British administration understood that the colony would not survive without the governmental system moving towards eventual full representation of the Chinese in the colony, leading to a much greater role for the Chinese in government services by 1980, and the formation of many Chinese advisory commissions and boards to link the government to the Chinese.

The real sticking point in the final decades of Hong Kong as a colony was the lack of democracy. Chris Patten attempted to address this in his term of office but by then it was too little too late, as the Joint Declaration between the UK and China did not explicitly guarantee these reforms.

Source: Tsang, S., A Modern History of Hong Kong, Welsh, F., Hong Kong: A History.

I'd just add to the excellent response above that Britain's attitude toward Inidia and Hong Kong were different. Hong Kong was an important financial and trading outpost. Very valuable but not Britain's highest priority.

India was the British Empire. Losing India seemed inconceivable for a long time. If they had to give away every other territory to keep India I believe they would. The Indian Mutiny in 1857 was treated like a personal affront to the British. And the crackdown was immediate. Britain took direct control of the colony after that (previously administered through the East India Company). And an entire civil service comprised of mostly British was established in India run by the government. (These reforms weren't all negative for Indians. They began to be allowed into the civil service in low positions and had access to some education but the overall affect was that of an occupying force.)

They also began using more British soldiers (largely Irish) in the army in India (which had previously used Indian soldiers, who began the revolt) especially in specialized units.

Hong Kong had a British appointed governor and a civil service but it didn't have anything close to the British administrative or military presence that was in India. They didn't have the same level of emotional investment in Hong Kong. And after the Indian Mutiny they were constantly on the look out for more uprisings in India. But, despite unrest, Hong Kong hadn't seen anything on that level. Their oppression in India was often part of a growing desperation to keep India under their thumb. And a growing class of British-Indians who saw India as their own.

(Their may also have been racial components involved where they saw the Chinese as slightly more advanced than the Indians although not quite to British level, of course. This sort of racial hierarchy is evident in the difference between India and African colonies where, for all of the very genuine oppression, Indians are definitely given more room to advance in the civil service, local politics, and education than black Africans. Of all people, Gandhi summarizes this thought process well.)

Sources: Rise and Fall of the British Empire by Lawrence James and Heaven's Progress by Jan Morris (I don't think Morris is the best source for facts but I think she nails the British attitudes and sentiment of the time better than anyone and British empire building was uniquely sentimental. Even when parts of her trilogy make me uncomfortable I think it's a valuable look at how the British saw their empire.)