If you imagine that the Time Lords on Gallifrey are like the Thatcher Goverment, then Faction Paradox would be the Punk movement. They are very keen on annoying authority, creating time paradoxes and being independent. They often do things for shock value because they find it fun, and investigate the edges of time technology, where it is dangerous to go.
In roleplaying terms - they are best as a neutral third party, that could help or hinder depending on whether you want to disrupt time or not. They tend to recruit humans but will also recruit other aliens, even though the founder of the group was a Time Lord. If Time Lords have damaged your history, your life or freedoms - then the Faction want you.
A lot of the Faction Paradox stuff is a bit of a mess, as Lawrence Miles wrote it into many of his Who books, then other people ran with it in a different direction, then I think the BBC tried to get rid of it.
Essentially they are a group of rogue time lords, who cause paradox intentionally to protect their own timelines. If you never existed, how can they mess with your past?
I highly recommend Alien Bodies, the novel that started it. Interference part 1 and 2 are another core source. They also crop up in Guns of Avalon and a few more. Then it all goes crazy with the Ancestor cell where they become a desease!
There is a full 'encyclopedia' called 'the book of the war' and a string of specifically Faction Paradox novels. The first is by Miles 'This town will never let us go' and is very good.
There is a comic, which is hard to find and only made it to 2 issues.
There are also 2 sets of audio dramas. The first (the 11 day empire) is a set of 5 continuing stories. The second set involve agents of Faction Paradox going up against the Osirans (pyramids of mars)
The whole thing is both rather cool and utterly bewildering. There is little cohesive cannon and a lot of people preparing for things that havn't happened yet and other time messing about!
I'd love to do a sourcebook for the rpg. :-)
Logic, my dear Zoe, simply allows one to be wrong with authority.
I like that even in the Lawrence stories you can see that the Faction isn't a united group, but their underlying love of individuality means there are splinter groups within the Faction with their own agendas and there isn't one over-all "dastardly plan". It gives more leeway for roleplaying purposes as well - it's possible to use them in many different ways because it could be a Godfather's pet project (Morloc's virus, or plans with Justine), it could be a single lone Faction member's ambitious or psychotic rampage (Father Kreiner or Cousin Shuncucker) or it could be heroic martyrdom for the greater good (Cousin Eliza).
The Pheno convention game I'll be running next year is a Faction Paradox game and will be adding to the existing mythos - ableit in a non-professional unofficial way. Nevertheless, any canon is good canon for the Faction
On a side note - I didn't really like "This Town Will Never Let Us Go" - I thought it was a bit too "try-hard", I liked his "Dead Romance" better. I generally prefer his EDA books and more than that - the audios.