TLDR: uninstalling 5.33 or upgrading to 5.34 (or newer) removes the malware as it is embedded within the CCleaner binary itself, even though thankfully it has been made inert since control of the remote servers is no longer in the wrong hands. Read on for more details.
Detailed analysis of the malware attack from TALOS
Blog post from Piriform about this
UPDATE:
In reviewing the Version History page on the CCleaner download site, it appears that the affected version (5.33) was released on August 15, 2017. On September 12, 2017 version 5.34 was released. The version containing the malicious payload (5.33) was being distributed between these dates. This version was signed using a valid certificate that was issued to Piriform Ltd by Symantec and is valid through 10/10/2018. Piriform was the company that Avast recently acquired and was the original company who developed the CCleaner software application.
Detailed analysis of the malware attack from TALOS
Blog post from Piriform about this
UPDATE:
So some vigilante apparently got into the command + control server the malware was talking to (detailed in the original tech writeup), and handed over the files there to Talos.
Talos then posted an update with more information about what the malware actually did, and some info about how many machines were affected:
http://blog.talosintelligence.com/2017/09/ccleaner-c2-concern.html
It looks like it was a highly targeted attack; the server was set up to check if the infected machine belonged to a predefined set of organizations, and then install further malware from another server. It's very possible the set of organizations changed over time.
Over 700k machines reported to the command and control server in a recent 4 day period, and of those, only 20-30 of them were actually instructed to install further malware. It's very likely that far more machines were infected than that, since the malware was in the CCleaner install binary for a few months.