Is it possible to work full time and do a part time phd in the area of software engineering in England saying that I want to use my job earnings to pay my tuition fees? And if I start a part time PhD funding myself this way then how difficult is it to get some funding from the same university (waiving the tuition fees at least) so I can quit job and focus totally on my PhD?
Answering the question as asked, the answer would be yes. There are part-time PhD positions available at many UK university Computer Science departments. There are currently part-time PhD students currently attached to many UK Computer Science programmes and the regulations and fees are arranged for both full-time and part-time students. However, that being said, the situation is far from being that simple, as indicated in the many comments.
The majority of part-time doctoral students are likely to be experienced industrial practitioners who are enhancing their qualification portfolio with a PhD or are working for a commercial research based organisation and have linkage between their work and a potential PhD. It is less likely, and indeed less acceptable that the student is working purely to fund the PhD. This will be considered by the university to be an unacceptable situation which does not give the student sufficient hours to perform the necessary work.
There are also further hurdles to this route. One is employment. Would an employer take on someone who is planning on a part-time PhD? Perhaps; but if you are not from the UK or the EU, then there is the issue of study visas and work-permits to contend with.
If it is finance, then as also mentioned in comments many departments have fully funded PhD research positions available and often advertise these in the academic press and online. The main database of these funded positions is hosted on behalf of UK universities at jobs.ac.uk and in that database there are 94 advertised PhD positions in Software Engineering as of today's date and four of those are advertised as being for part-time students.