Electronic – DIY ATX PSU turns off when attaching load on 12V rail

It should be possible and not too complicated. There are many guides explaining the process. Here's one.

UPDATE:
To complement the other answer:
Some computer PSUs may already have needed load installed and can provide stable voltages out of the box, but you can't rely on that.

The instructable recommends use of $10\mbox{ }\Omega\mbox{, 10 W}$ resistors for $\mbox{+5 V, +12 V}$ and $\mbox{-12 V}$. The $\mbox{12 V}$ resistors are going to be dissipating around $\mbox{14,4 W}$, so my recommendation is to get some bigger resistors, just in case.

The other option would be to use resistors with higher resistance for the 12 V lines, for example $22\mbox{ }\Omega$. They'll use less current, but I think that it won't be a major problem because the screen will use considerable amount of power too.

This isn't mentioned in the instructable, but I'd put some $10\mbox{ }\Omega$ or $22\mbox{ }\Omega$ resistors on the $\mbox{+3.3 V}$ line too.

Another part that has been omitted is how to locate the power supply rails inside the power supply itself.

Here you can find the color code for computer supply wires. Just follow the wires inside the case of the supply and see where they end. The wires you're interested in are yellow for $\mbox{+12 V}$, orange for $\mbox{+3.3 V}$ and red for $\mbox{+5 V}$.

Don't forget the LEDs for gray and purple wires. You can use tool like http://ledcalc.com/ to get the correct resistor for the LED.

The PSU's output feedback is based on the +5V rail. You need a load connected always to that rail. I suggest you leave the +12V rail without load (if it works).

^{simulate this circuit – Schematic created using CircuitLab}