PC power supply charge 12v battery
No, it will almost certainly not work. A computer requires both 12V (for fans, hard drives...) and 5V (for motherboard...). Both these rails have to have a common ground. So, a PC power supply has all rails at a common ground. Do not connect any rails in series (or parallel) or it will destroy the power supply (possibly with smoke and loud bang).

With two separate power supplies, it may or may not work. Most PC supplies have the common ground also connected to mains earth. Disabling that may work. However, one has to be careful. Simply disabling mains earth is both risky and may be futile. The mains neutral may be connected to mains earth outside the house, and connected to chassis (the metal box) via a capacitor. One actually has to disconnect the low voltage common connection from chassis (which is usually difficult due to the way the PCB is constructed in most supplies). Typically, insulating washers are used (search on the web for intructions). However, the silver lining is that, this needs to be done on only one of the supplies. Make sure you do not disconnect the neutral to chassis capacitor (if there is any), and the chassis to mains earth connection. Make sure no part of the battery is touching mains earth (through acid leakage or otherwise).

Above all, be safe when working inside an SMPS. Discharge primary capacitors with a resistor on a wooden stick.