I want to install Chrome on my Kindle Fire HD.
The APK file is available at third party sites but I don't know which are trustworthy. Any suggestions? (I'd love to download it from Google but don't see any download links from there).
I cannot comment on how reputable they are - I've not had any issues but your milage may vary - but I was able to install Chrome on my Kindle Fire HD via the http://1mobile.com marketplace.
You have to register and install their client on your Kindle, but then it works like the built-in appstore.
I'd use it for free apps (and check the permissions any app asks for).
On the whole I've not found much improvement using Chrome vs Silk (partly because some of the Chrome functionality requires Google Services - eg for login - and that's not present) but if you need it then it seems to be a good way to go, and it keep it up to date as new releases are made available