As Hotei said, my first suggestion would always be that if your laptop has an alternate video output, you can always try using that... However, in the past year or so, this solution has worked less and less - in fact, not at all on recent models (where the actual GPU is dead).
You are sort of in luck that the computer actually boots (some of the time, faulty GPU causes the computer or the OS to fail booting)... If you are happy with it like that at the moment, I would stick with it.
I personally have one laptop like this that a client gave to me as part-payment and I use it for testing purposes. (Recently, a little hobby of mine is PXE booting Linux boxes and starting a web server with a control panel!)... You could always look at PXE installing OS's, but it isn't that easy to get right.
If you want to use this computer again, and it is under warranty or you have an Nvidia 8 series card, it may be worth speaking to your manufacturer - In the case of Nvidia, many have extended warranties due to heat problems and as for manufacturer - it may just be worth trying your luck!
If you do not have any luck with the above, it may be worth getting a cost from your local IT Specialist (Going to a professional is not admitting defeat!)- I do not know how they do it in Amsterdam (profile!) but in the UK a few shops do repairs themselves, others send laptops to a third party for component repair. You can usually get a BGP reflow for under £150 that should correct "most" problems... but you may be pleasantly surprised by the fee.