Run a launchd job after Time Machine backup
A simple solution is to use the tmutil command to manually trigger the backup and follow it up with your process (as part of a shell script that does both these steps).


First, turn off automatic backups from System Preferences > Time Machine or using the command tmutil disable
Use tmutil startbackup --auto --block to trigger a backup once every hour (the --block option is required to make the execution wait until the backup is finished)
Follow this with launching your process


See man tmutil for more information on using Time Machine from the command line.