Crazy Jamie gives good advice and I hope he/she will not mind me adding to it. It is important to emphasise that there is no right or wrong path. Each set is different, and although the process is often arbitrary you would be wise to do your homework thoroughly. For instance, Crazy Jamie draws a distinction between Manchester sets being less concerned about a local connection than Newcastle. I do not practice in either location, but my knowledge of at least two Manchester sets is that their members often have a local connection, be that schooling, university, and/or the BPTC. That is not a criticism of Jamie's very good advice, it is just important to do your research and tailor your applications to each respective set.
Remember the aim is to draft an application that will land you an interview. Know the set that you are applying to and sell yourself to it. That means knowing the profile of people they take on; knowing the main practice areas (not ones that are on the website because they did a case 5 years ago); talking to people who have applied there and interviewed before for tips, and if you know or have the opportunity to meet members of Chambers make the most of it. Chambers will inevitably consider a mix of things including your academics, experience, your written advocacy on the application form, and whether the reader thinks you could be a "good fit". In my set commitment to Chambers/the local area is important and should be obvious to anyone who has done their research. It may seem an obvious thing to say, but do not give the reader an easy excuse to dismiss your application.
Returning to the application form. I appreciate how much of a pain they are to complete and how utterly ridiculous and vacuous many of the questions are but ultimately it is a piece of written advocacy: do not fall into the trap of filling out the form as if you are filling out a survey with a chance to win a £20 voucher because you are so fed up; do not think that it should be obvious that you are good enough to be interviewed because of the university you went to and the experience you have; do not think sucking up will be sufficient. Answering the questions is a golden opportunity to show the reader that you are engaging and persuasive in writing. True, you can feel like you are coming across as an arse when answering these questions (and there is a set for every personality at the Bar...) but if you treat the process as akin to drafting written submissions then you should get the balance right.
On BPTC choices, in my experience the course and the providers are not generally held in high regard so where you choose is not likely to make or break you. True, there are wildly different success rates but that may be due to many factors such as the strength of the cohort (what they have done before the BPTC), where people have applied etc. I can only speculate, but I would assume that a London set is less bothered about where you did the BPTC than a regional set. So if you are going to throw applications around you may consider choosing a regional provider to be the better option. I did the BPTC in London but had a cheaper alternative option. If I was choosing again I would save a few thousand pounds and not do it in London.
The BPTC choice has the potential to be a red-herring. As with this entire process you have to look yourself in the mirror and be honest and realistic, the more so in a shrinking profession. Look at the most junior tenants in the sets you want to apply to: do you match up? Getting pupillage is competitive and if you want to go to the Bar you are probably quite competitive too, but do not let the competition blind you.