How do I update my jailbroken Apple TV to 5.3? I have a jailbroken Apple TV (black), which has firecore's aTV flash black installed.

How do I get it to update to the latest ios so I can use HBOGo?

Settings > General > Update just says "You are up to date", yet it's on version 5.0.2.

aTV flash updated to 2.3 without issue.
If you are doing over the air(thru apple tv) update, you may have installed a hack that disables automatic updates so you won't lose the jailbreak. 
Connect to your computer and do a DFU update or uninstall the hack that disables the update from within apple tv/nito or where ever it's installed and try updating again. Although I recommend DFU upgrade.