Try AMD's Raptr it's free to download and works for AMD GPU's. Not sure about your AMD GPU though.
I have a Hauppauge HD PVR2, which I use for my Xbox 360 and Xbox One gameplays, and it can record my PC gameplays too. Not that with AMD Raptr, sometimes when you record videos with some games, it doesn't record properly and the end result when you try to play the recorded video, is just a black screen. Shadowplay I think is only for Nvidia GPU's. I wouldn't recommend getting free screen recorders etc as they may have low frame rate recording or be spamming you with ads/spyware etc and will not be as good as paid software/hardware capture material.
My PC has an AMD Radeon R9 390 which tonnes of sources on the web state is a High End GPU. It also has a 6th gen Intel Core i7-6700k quad core processor and 16GB DDR4 2400Mhz HyperX RAM and a Kingston Fury HyperX SSD. I have tried Fraps, didn't like it. I even think AMD's Raptr UI is cleaner than Fraps.