Watch out, while solid state relays are triggered by DC, much of the time they only switch AC as it rely's on the current becoming 0 to stop the circuit.
The screen will probably last longer if you power it from the 12 volt rail of the PC's power supply.
The amplifier will probably have 3 power inputs.
The positive and negative need to be wired back to the battery. The remote/turn on can just be attached to the power supply's amp turn on output that you were planning to attached the solid state relays to.