Brichester is a fictional place by horror writer Ramsey Campbell, so you can pronounce it however you want.
My first guesses are Brickstur, Bristur and Brichstur, but the ch in -chester names are usually pronounced (like Chichester, Manchester, Dorchester, Chester), unlike the hard k sound in -caster names (Lancaster, Doncaster, Tadcaster) and the s sound in -cester names (Leicester, Cirencester, Bicester).
Someone on RPGnet says it's pronounced:
Brychester - with the "i" pronounced "eye"
like in "eyeballs turn to mush as you look
at the thing from beyond reality"
So either that or Brichestur (a bit like Bridge-chestur).
Alternatively, you could find an interview with the author and see what he says. He's English, from Liverpool, and the place is set somewhere in Gloucestershire (which, incidentally, is pronounced something like Glosturshur).