The problem is taking the Game Boy video output and converting it to VGA or DVI. It can be done, but it's much more complicated than simply using a SNES with a Super Game Boy.
The good news is, someone else already did all the hard work:
As for the carts, you absolutely need to have a scaled PCB with "pins" on the giant carts, with wires inside the cart to go to a real cart housed inside. At that scale, I'm guessing a hand-made edge connector for the giant Game Boy will work fine and be relatively easy to do.
edit: I just read the "Intercepting the Gameboy LCD" page and it's quite easy to do if you have something fast enough to capture the data. Converting that data into a VGA or DVI signal is another story but I think there's boards out there to do exactly that.
The last problem will probably be about scaling the image properly so that it doesn't look like washed-out crap. You do want clean, sharp, big square pixels on that giant display, right?
The Game Boy is/was 160x144 pixels, so you have a choice between the following resolutions if you want perfect scaling:
- 160x144
- 320x288
- 480x432
- 640x576
- 800x720
- 960x864
- 1120x1008
- 1280x1152
It's a perfect match on the horizontal scale for normal VGA/SVGA/etc resolutions but it's another story for the vertical scale. For perfect scaling you will need to use a resolution that's one-scale behind the native resolution of the display (ex: 1120x1008 on a 1280x1024 display). That means a border around the Game Boy image but the pixels on the real console didn't go completely to the edge either, so it's still "perfect" when you think about it. What display are you going to use? Since you have already cut the LCD area I'm guessing you already have one ready to use and the scale of your giant Game Boy is probably based on that display, so what size is it, what is the resolution and which inputs does it have?
edit 2: added templates of the native LCD resolutions vs the maximum perfect scaling of 160x144 that can fit inside each.
edit 3: removed the black/grey images since I posted new ones with Zelda screenshot a few posts below.