As a self-funded part-time doctoral student, I feel quite qualified to comment on this one!
All the above advice is excellent so I'm just going to add some further thoughts:
If you're doing a part-time PhD around a substantial job, your university experience is quite different. You enter that strange world of studying at evenings and weekends, at times when the university library is often quieter and frequently shut (and I'd say that Cambridge is pretty rubbish for out-of-hours libraries, check the opening times for the library of your subject or the UL & you'll see what I mean - I miss the 24-hour library of my Masters university a LOT). Also the cafes close just when you need that extra coffee to keep going.
Self-funded doctoral students develop a certain mentality which can't be avoided. The temptation to work out your cost per contact hour is sometimes irresistible, especially in non-science subjects: 'I'm paying thousands per year for this and I've only seen my supervisor twice this term…..' This is, of course, a rubbish argument, because you're getting lots for your money but I reckon every self-funded student says or thinks this on occasion.
Part time students can get forgotten, as gutenberg says. I'm lucky in that my department is extremely proactive, with an excellent course leader and administrator, but it's not the same for every subject.
In my subject (a social science) there is limited funding anyway, so absolutely no stigma attached to self-funding. If you make a name for yourself during the doctorate, with networking and publications, there will be post-doc opportunities. Don't know how much this applies to other subjects.
There is anecdotal evidence (there's probably actual evidence but I wouldn't know where to find it) that self-funding students are less likely to finish than funded students.
Part-time doctoral students definitely have limited funding opportunities - some of my colleagues receive part-funding from their employer but otherwise there's not much available.
Would I advise people to do a part-time doctorate? I enjoy it. The pressure is the same but diluted (so if external factors cause you to have a bad term, you can make up for it the following term). Having a job in the 'real world' helps you keep a sense of perspective and helps with money! It also allows you more time to develop your thinking and test your theories.
Would I advise someone to self-fund? - probably not, actually, not until they'd explored every option for funding over a couple of application cycles if necessary. This isn't because of any effect on their future career opportunities, it's more that I wouldn't encourage anyone to take on a financial commitment of that size if there's a chance that someone else could pay the bill for them..