What is the best way to clean your clothes dryer exhaust?
You can probably do it yourself cheaper than hiring it out. You could try something like this vent cleaning system. It seems to have fairly good reviews on Amazon, and for $25 it's worth a shot. It claims to have a 12' reach and the ability to navigate turns in the vent, so you may have to go at the last 3' from the other end. 

Note: You'll need a drill to attach it to. A shopvac would also be helpful, but if you don't have one, the kit includes an attachment to use with your dryer.