Despite its promises, Sony isn't doing a great job of getting PSOne games over to the Xperia Play store and there's only so many levels of Crash Bandicoot most of us can stomach.
1. If you don't already own it, go out and buy a second hand copy of the game you want to play, most of them are a $/£/€ on eBay, and we wouldn't want to encourage the kind of lawlessness that might end civilisation. (Elite Rating: Harmless) You can rip your own ROM file from the disc using commonly available tools, or for speed, download the ROM using an Android app (below).
2. Using your Xperia Play search for and download the following apps: RomFetcher: Playstation and FPse for Android, both cost just a few $/£/4€. (Elite Rating: Mostly Harmless)
3. When using your Wi-Fi connection,use the RomFetcher app, search for the game ROM that you have brought and wish to download, newer games are a lot bigger, so if you're after something speedy aim for an earlier PSOne title. (Elite Rating: Naughty Boy)
4. Now launch FPse and find the file you downloaded, it will usually be stored in "SD Card, ROMS, PSX, game_name.bin
6. When the game has loaded, go to the Settings menu in FPse, change the Screen Size setting to "Original" and on the Input screen, change the pad type to "Analog." Assign the hardware keys to your controller and you're ready to go.
Just as an example, here's a demo of someone playing Tekken, if you have trouble with performance then give more resources/priority to your FPse in Android Settings.