The PS4 scene has had a bumpy start but this year is a different story. It all started with the release of the kernel exploit for FW 4.05/4.55 and then more application started to arise like The Homebrew enabler and now the ability for the PlayStation 4 to play PS2 backups or others call them ISO images.
The reason it can play these backups is the PS4 can already play PS2 games, they just need a little tweaking. It’s just like the PS3 with CFW in it where we can play a lot more PS2 classic and even PSP games too as these are features of the system, so that’s why the PS4 can do PS2 backups. Over the weekend developer flatz mentioned that the PS4 could play PS2 ISO images by building a custom PKG file..
https://twitter.com/flat_z/status/954743398082203648
Youtuber Andrew2007 demonstrates the PS4 on FW4.05 running a PS2 game. He shows it playing DragonBallz Budokai Tenkaichi 3 and its running flawlessly on the PS4, ps2 emulator and even games like Klonoa 2 work on the PS4 where it didn’t on the PS3, ps2 emulator. 
If you have a PS4 on FW4.05/4.55 and would like to do this you are going to need to do a little extra digging but here’s are a few places you might want to check to help start making those custom PKG files.