There are several ways to play game backups on the original PlayStation. One common method involves swapping a genuine disc for a backup disc while the system is running. Another method is to install a modchip that allows the PlayStation to boot directly to backups. This is a guide for PSone MM3 modchip installation.
An MM3 modchip will allow you to play your game backups, as well as games from other regions. You can get an MM3 chip for fairly cheap at around $4-10, I got mine from a US seller on eBay preprogrammed and prewired for $8 shipped.