My Slackware64-current and Catalyst 10.9 plays just fine, I have a Radeon HD 5750 card. What do you mean by after building packages another installer launches? I usually built my ATi packages on init 3 (console only) like:
# telinit 3
# ./ati-driver-installer-10-9-x86.x86_64.run --buildpkg Slackware/All
After the two process above, I install the created packages, load the fglrx kernel module, and proceeed to init 4 again (or reboot).
# installpkg fglrx*.tgz
# modprobe fglrx
# telinit 4
If you said about X being slow, check your /etc/X11/xorg.conf file. Below is an example from my xorg.conf file, make sure you set Driver to fglrx as below.
Identifier "Radeon HD 5750"