I would upgrade to the latest 1.7.x, then latest 1.8.x, then latest 1.9.x. Make sure to backup the database before any upgrades and read any changelogs between versions.
You will also want to make sure the customer has not modified the core files in Magento. If so, those modifications will need to be implemented into custom modules.
After each upgrade you'll want to do very thorough testing to verify things still work. Test as many use cases as possible in as many browsers as possible. After you're done with the upgrades test again to verify everything is still working.
Keep in mind that in 1.9.3, Magento's Adobe Flash based uploader was replaced with an HTML5 based uploader. Please make sure that any custom functionality does not depend on this.