Fly to Sofia from various UK airpports as follows:
Wizzair from Luton, daily schedules.
EasyJet from Gatwick, daily schedules.
EasyJet fly from Manchester, twice weekly.
Air Bulgaria from both Heathrow and Gatwick.

Transfer from Sofia to Bansko:
Minibus: £25 - £40 per person depending on group size. Many minibus companies operate including:
http://www.banskoexpress.com
Hire car is a 2.5 hour drive: http://www.thrifty.com
Taxi: Approx £150 total return for group of 4
http://mitch-transferbg.com/