Depending on which version of Pilot Pro you are currently using, there is a slightly different upgrade path to version 5.
From Version 4
In the version 4 app, go to the Settings tab and then hit Push Logbook. This may take several minutes to complete. Once this is done, install version 5 and your logbook data should start syncing automatically.
iCloud is required for sync with version 5.
Note: You cannot sync from version 5 back to version 4. It's one-way from 4 to 5, and only when you use the Push Logbook option.
From Version 3
Hopefully by now you have connected version 3 to Dropbox and have a Logbook.pilotpro file in your Dropbox account. This is the file needed to upgrade to version 5.
In the version 5 app, go to Tools → Import and look for the section where you can browse for your version 3 logbook file.