Yeah I know that's pretty rough. But the point of failure occurs at the 'motherboard runs everything stage'. POST is an pre BIOS test system it checks for errors occuring when loading the BIOS.
For the BIOS to load you do not require a hardisk or graphics card; however your CPU, motherboard and (the crucial part of) the RAM must function properly. If it's the RAM you are lucky.
Otherwise you are in for as good as a computer replacement (what with hourly wages and stuff to get things repaired/replaced.) Especially considering you have a laptop.
Since you mention a pink/purple/green screen and other general signs of overheated GFX cards.. I hope the GFX card is actually a seperate GFX card and not a chip attached onto the motherboard? The latter case may explain everything though: if that component had overheated it would've destroyed part of the motherboard fairly easily.
You can try if it really is your lucky day by getting a free, low-requirements & basic OS which also runs from a live-cd; burn that (the live-cd image) on a CD/DVD; mount it in the CD/DVD-drive and try if you can get till POST on the broken machine. When there try to get your machine to boot off the CD/DVD drive by hitting the key which makes your BIOS switch to the screen that lets you select your booting device. Select the cd/dvd rom there.
Now if it is your lucky day the lightweight OS should boot off the CD/DVD and you can at least install it. That means you can at least recover a working machine under a different OS; which you may use to figure out what parts of your PC are actually broken...
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