The steps should work for other keyboards as well. I've gotten it to work with Fleksy and Google Keyboard. YMMV.
1) Download and install Swiftkey to your HDX through Google Play Store (if you have hacked it onto your device), Amazon App Store, Aptoide or 1Mobile, or even sideload it. However you do it, install it.
2) Once you got it installed, run the Swiftkey app, which is also the installer. Keep going as far as you can until it tries to trigger the settings and so on. It probably will bring up Keyboard Settings, and you'll be utterly lost here.
3) Bring down notification, and go into settings (far right button, usually)
4) Go down to Language & Keyboard and tap it
5) Tap on "Keyboard Language" (should be 3rd one down)
6) Keep scrolling down the languages (and you'll see TONS) until you finally find Swiftkey, make sure it has a checkmark (if not, add it now).
Optional: Also enable any other keyboard you may have loaded. Beware that some keyboards appear here actually as a language. For example, Google Keyboard appears as "English (US)" if you have that installed, though Fleksy appears as itself.
7) Once you have checked all the keyboards you want to enable, scroll back to top
8) Tap "Default language", which is first choice, and you should get a pop-up of all the different keyboards/languages you enabled, and among them should be Swiftkey (or Flesky or whatever keyboard you put a check mark on in step 6) Pick your favorite keyboard now.
9) Hit BACK to go back to "Language & Keyboard" screen
10) Tap "Keyboard Settings" (4th choice?) If you had previously selected Swiftkey, this should bring up Swiftkey Settings, or perhaps the installer if you have not completed the steps like Sign into Google Plus to enable CloudSync. Complete any remaining steps in the installer.
Your new soft keyboard is now active! Enjoy!