How do I ensure computer stays awake for Back to My Mac?
The obvious solution would be to set the Mac never to sleep in Energy Saver Preference Pane. If your home router is an Apple Product, you could enable the bonjour sleep proxy setting on your Mac that's labeled "Wake for Wi-Fi network access" and even a sleeping Mac will be woken when Airport detects the remote control packets coming in for your Mac.

Alternatively, you could use a free program called Caffeine to prevent sleep. Lastly, consider Edovia's free screens connect control panel to prevent your Mac from sleeping.


http://edovia.com/en/screens.html


Scroll to the bottom of the page and download their control panel installer. You would need to make an account with them and then run that control panel - choosing the "prevent your Mac from sleeping" option.

I switch back and forth between using their location service and Apple's and have never had the Edovia software conflict with Apple's or cause any issues.