I have the same system. I've gone through multiple downloads of Radeon Crimson software. The most confusing thing has been versions bouncing. Today (April 15, 2016) the Catalyst Control Center recommended I download new software - Catalyst 15.7.1 without Dot Net 4.5 for Win 8.1 64 bit. The current video driver version is 15.301.1801.1001 dated 1/30/2016. I'm about to reboot and figure out where this revision heads next. I have noticed improvements in graphics rendering in several games. It is worth noting that the HD6310 is considered legacy hardware and there are vast changes that have happened since the chip set was introduced in 2010. I've tried to play Neverwinter and the painting in of textures is seriously slow... that is probably due to the AMD E-300 chipset running at 1.3ghz. I've set graphics on lowest settings to see how it responds and the CPU is usually clocking at 100% and there's lag present. Doing anything in party formed in game usually creates additional lag to the point that the game becomes unplayable since the character runs into walls or walks off cliffs.
Resulting versions: Quick Stream - 220.127.116.11; AMD Gaming Evolved 2.1.000
January 2016 I downloaded WHQL Radeon Software Crimson Edition 16.1 64 bit Win10 Win8.1 Win7 Jan7. I've also tried the Cromson 15.12 Win8.1 64bit version in December 2015. The sequence numbers on these updates gets confusing. I realize though that CCC can have one version, graphics drivers another, and Gaming Evolved still another.