My question is in the title; in what ways were China's economy 'liberalised' and in what ways did this allow it to become a model for the liberalisation of previously planned economies while preserving the political class in Asia (a la Vietnam)? How did these reforms improve the quality of life of those within China?

Previously, much of China's economy was run by state-owned industries. People living in the cities were assigned jobs when they got out of school, with the promise of job security and a stable wage. Peasants were to grow a quota of crops that were to be sold to the government at a certain rate. Peasants and urban residents were denied the freedom of mobility with the hukou system, although it was easier to go from the cities to the countryside compared to the reverse.

This changed with the economic reforms of the late 1970s and 1980s.

With the new economic reforms, China allowed for the existence of a limited private sector. While China kept a hold of numerous state-run factories (namely in heavy industry and telecommunications), the country has also allowed for limited competition to rise. It also allowed light industry to develop, first in the economic zone in Shenzhen (located in Guangdong Provence, right outside Hong Kong), and then all over the rest of China. Factory owners had the freedom to lay off workers, thereby weakening the "iron rice bowl" (aka the promise of job security). Labor contract reform allowed more workers to be hired temporarily at factories. Peasants were allowed to sell excess crop to private markets in the villages, as the "family contract system" was implemented. By the 1990s, peasants were free to move to the cities to work in the factories, leading to the rise of a huge migrant population that would work in the factories and send money back to their families in the rural villages. This was the beginning of "socialism with Chinese characteristics".

With these economic reforms, the standard of living rose quickly. An emerging middle class with their wages purchased consumer goods such as makeup and fashionable clothes. Agricultural and industrial production both rose. State-run Chinese industries are traded every single day on the stock market, and foreign investors have come in to China in order to build factories and make cheaply made goods.

Even so, not everything is well. Although the standard of living has risen, the gap between the rich and the poor is widening. Workers working contract labor, and women as a whole have faced discrimination from employers. The liberalization of the economy undoubtedly had political effects as well, as people began wondering if the Chinese Communist Party would give other freedoms as well (like political freedoms). Well, they didn't. Instead, they cracked down on the Tienanmen Square protests on 4 June 1989, causing a slowdown onto economic reforms. Eventually, after Deng Xiaopeng made a visit to Shenzhen in 1992, the economic reforms started up again, but much more cautiously, and ever since then the CCP has cracked down and kept a tighter grip on businesses and political freedom (much of it which goes outside the scope of this subreddit as it violates the 20-year-rule, so I'll stop here).

Sources for this post:

(Note: Some of the following sources are PDFs from JSTOR. If you have JSTOR access to these articles, whether through an institution such as a university, the public library (sometimes they do that!), or from a free MyJSTOR account, I highly recommend them. If you have issues with access, please feel free to PM me.)

The Party: The Secret World of China's Communist Rulers by Richard McGregor

"Smashing the Iron Rice Bowl" by Niel C. Huges (http://www.jstor.org/stable/20048966)

"Socialism, Capitalism, and Class Struggle: The Political Economy of Modern China" by Minqi Li (http://www.jstor.org/stable/40278337) (I would be careful with this source, as the article has a really left-leaning bias, to the point that it subtly hints at possible Great Chinese Famine denial.)

"Rice Bowls and Job Security: The Urban Contract Labor System" by Pat Howard (http://www.jstor.org/stable/2158633)

Additional reading:

Personal Voices: Chinese Women in the 1980s by Emily Honig and Gail Hershatter

Factory Girls: From Village to City in a Changing China by Leslie T. Chang (this one kind of violates the 20-year-rule a little, to be honest, but I still highly recommend it)

Also, if anybody has any books to recommend on post-Mao china I would really appreciate it, I'm finding it a little difficult to find one that focuses on the economic reforms of the late 70's.

There was not much economic reforms of the late 70s really, Mao died in 1976, Deng only took power in the 78-79 period. Reform was mostly a 80s-90s thing.

My question is in the title; in what ways were China's economy 'liberalised' and in what ways did this allow it to become a model for the liberalisation of previously planned economies while preserving the political class in Asia (a la Vietnam)? How did these reforms improve the quality of life of those within China?

Decommunization of agriculture is the best example of early reform that worked, the creation of town-village enterprises (semi-private) is another example of building industry outside of the inefficient planning apparatus.

If you have specific questions I can answer them, but the topic is rather exhaustive. If you want a good read on the reform process then Barry Naughton's The Chinese Economy: Transitions and Growth is a pretty good read.