How did the government of South Africa deal with apartheid?

If you are referring to how the South African government negotiated the end of apartheid, I would argue that a series of reforms under Prime Minister F. W. de Klerk made the change possible.
However, before we discuss how the South African government dealt with apartheid, we first need to explore how apartheid came into existence. It was the South African government that crafted the apartheid laws.
In 1948, the Afrikaner National Party won the general election and immediately put a series of laws into place. The purpose was to protect the rights of minority whites. Essentially, the white population of South Africa feared the prospect of losing power and influence to the majority black population. Here are some of the laws the Afrikaner National Party championed and brought into existence:
1) The Group Areas Act of 1950: According to this law, black citizens could only live in certain regions. Segregation was forced upon the populace and strongly enforced. Millions of black South Africans became displaced and suffered greatly.
2) The Population Registration Act of 1950: According to this law, people had to register with the government according to their racial group. Registering in such a way ensured that black populations would almost certainly be discriminated against in terms of job, education, and housing opportunities.
3) The Prohibition of Mixed Marriages Act of 1949: According to this law, South African citizens of European ancestry were prohibited from marrying citizens of non-European ancestry.
There were, of course, several more egregious laws that reinforced the practice of apartheid from 1948 to 1994. During this time, opposition political groups such as the African National Congress and the Pan-Africanist Congress went underground. Such groups were banned by the Afrikaner Nationalist Party in power. At the time, Nelson Mandela was the leader of the African National Congress.
In 1986, South African President P. W. Botha tried to negotiate a truce with Nelson Mandela (who was in prison). However, Mandela was unhappy with the terms of the truce and elected to remain in prison. When the new Prime Minister (F.W. de Klerk) took office in 1989, he began a series of negotiations with Mandela.
In 1990, de Klerk announced that he was releasing Mandela from prison. De Klerk also lifted the ban on the African National Congress and other opposition parties. On February 11, 1990, Mandela was freed after 27 long years in prison.
On July 2, 1993, de Klerk held South Africa's first democratic election in 40 years. As a result, Mandela was elected president of South Africa's first postapartheid government. In that same year, both Mandela and de Klerk won the Nobel Peace Prize. For more information, please refer to the links below.
https://www.blackpast.org/global-african-history/apartheid-1948-1994/

https://www.pri.org/stories/2013-12-05/nelson-mandela-man-who-brought-south-africa-out-apartheid-dies-95

https://www.thoughtco.com/when-did-apartheid-end-43456