I'm not one of those people though, so here are my tips:
Measure the current draw of the inverter when switched off. Some inverters draw several hundred mA when off, and that can mean a discharged battery in the morning and a deceased battery after a few dozen such events. You may find that you need to add a separate power switch to completely turn it off.
Keep the ac power cable short and straight, and be mindful of where it is. It is more than happy to induce a 60hz buzz into your audio if you let it. Prepare yourself to use a ground loop isolator if it becomes necessary, even if you consider it a bandaid as may do. That inverter creates a rather large EM field and if you have a ground loop between any of your equipment which would otherwise be irrelevant, adding an inverter to feed it may require you to do something about it.
If you were hoping to use S3 stanby, stop hoping. The inefficiency of the inverter plus the atx power supply will amount to many times more than the power drawn by the computer itself and will drain even a yellowtop in just a few hours. Unless you rig a linear 5v regulator for standby power, hibernation and cold booting are your only options. Also, if you are considering a separate 5v standby supply, be wary of older motherboards. Power saving methods were horrid - if even present at all - on older hardware.
Lastly, it IS entirely possible to have a noise-free carpc setup using an inverter. No matter what anybody tells you, it's possible.
EDIT: And I don't mean to give advice that you didn't ask for, but I just want to make sure you're aware of the difference between vga resolution and rca resolution. If your monitor doesn't have vga input, your eyes will be in a world of hurt if you try to view normal text outside of a frontend.