As Hanso mentioned tuneECU is the way to do it, which is super easy to use. You just need a compatible cable and to download the software and drivers.
here is a link to the cable many people have: Lonelec (look for the triumph tuneecu cable)
As for how to use TuneECU there are lots of threads in the trading tunes forums that talk through it and give great advice from people far smarter than I.
factory rev limit would be 6300. might possibly be mixing it up with the speed limiter which would stop you revving past 5200rpm or so in 5th. unless previous owner if any might have changed speed limit to say 150mph which would be around 5700rpm in 5th