I wanted to do something similar for my own arcade.
My solution was pretty straightforward, though I can think of a couple additional solutions.
My solution was to use a master controlled power switch, so when you turn on one device, that turns on additional devices (marquee, other lights, stereo, television, etc). Something similar to
This Device. I put the computer on the master control, and use a momentary switch mounted at the top of the arcade to turn on the computer, and when the computer turns on everything else turns on.
That only works for my particular set up because my monitor does not have a soft-on, it has a mechanical switch. If your TV has a soft power on then you will need a different method.
If you have a soft power-on TV, you might first check the service menu to see if you can enable the television to automatically come on when power is applied.
If not, you might be able to use a ganged momentary switch (SPDT), then run wires to both the power button on your computer and to the power button on the television, then the single switch should turn on both devices. Using the same master controlled power outlet I mentioned earlier, and you could have either the TV or the computer plugged into the master and the other device plugged into one of the unswitched outlets, then have all your peripheral devices plugged into the switched outlets (audio, lights, marquee, etc).
A separate solution, though much more technically demanding, might be to bypass the softlatch power control circuit for the television, though I don't know if that would be possible on all sets, especially if its controlled through an MCU. If, however, it uses a Solid State relay or discrete components it should be pretty trivial to jumper the output.