How could one best go about research in Pure Mathematics, whilst holding a full/part-time job?
I have published a little work in number theory while working full time. There is no big secret, just read about the problems that are at a level you can solve, solve the problems and publish the results. 

Publishing in a journal as an unaffiliated researcher is harder than for professional academics but not impossible. Learning TeX will help because many journals want LaTex submissions. Most background information is available online if you search for it. You can join polymath projects, use mathoverflow etc 

There are some advantages of working outside the system. You dont have to go around pleasing the people who might offer positions and you can work on what you want rather than what a math department or funding agency wants. So enjoy it as a hobby while earning a proper salary.

Update: I will add a little more specific advice. If you were doing a doctorate in a university you would have a wise old supervisor who would make sure you are heading in the right direction. If you want to work outside academia there is a danger of getting on a track that has no hope of success. Do not fall into the trap of thinking you can tackle a well known unsolved problem directly. This is a mistake very frequently made by outsiders. 

It is better to start on simpler problems that you can handle and use them to build up expertise even if they are not of much interest to others. One way to find such problems is to start from a well-known one but then generalise it in some novel way to a bigger class of problems. Then look for special cases that can be solved and then generalise your solutions to solve a wider class of cases. 

Finally, use your skills as a programmer to find solutions of interesting problems and then look for patterns. When you find patterns, make conjectures from them that you may then be able to prove. Try solving the problems at https://projecteuler.net/ to perfect your algorithm skills