I do it a mix of both Thoth and Smoka's ways.

1) Download the new driver and save it someplace handy, like the desktop.
2) Remove the old driver. I usually do this by going to the Change/Remove control panel and then launching the AMD Catalyst Install Manager and having it uninstall the driver software. I don't use a program like Driver Sweeper or anything like that afterwards; never had the need. Also, if you're using a program that hooks into your drivers like MSI Afterburner...shut it down and make sure it won't try to boot with Windows. (Temporarily at least; I use MSI Afterburner for a fan profile myself, and I suspect it wouldn't react well to searching for GPU drivers that have been uninstalled. So I disable the boot with Windows feature and shut it down until the uninstall/install dance is complete.)
3) I do reboot when prompted by the uninstaller...my system apparently being content to derp around with whatever video drivers are on my Intel motherboard (something like 800x600 resolution and no Windows Aero

) until otherwise instructed. It doesn't revert back to an older driver version like Thoth's does.
4) Run the installer for the new drivers. Twiddle thumbs. Reboot when prompted. Resolution, Aero settings, CCC settings etc. all snap back to what they were with the old drivers at this point, and you should be good to go.

(Also, restart MSI Afterburner or whatever at this point, if you use them.)