Is Soul Calibur V available on PS4?

Is there any way of getting Soul Calibur V on PS4? Or do I have to buy the PS3 game and then pay to convert it (how would that work)?

No, Soul Caliber V is not available on the PS4, only the PS3. You can't typically transfer games from PS3 to PS4, but in some cases buying one version entitles you to another console's version (these will be marked with a cross-platform/cross-buy icon and mention it in the description somewhere). In this case, there just isn't a PS4 port of the game, so there isn't a way to play it on PS4.

It's worth mentioning that Soul Caliber V was announced for the PS Now service in mid-2016, but it was for Japan only.