In alot of cases a black screen is a graphics problem. Sometimes that can be remedied by editing the kernel line with the proper command.

Did the bootloader (Grub) install correctly during the installation?

Since you installed Tiny Core w/o a GUI and cmd-line only it may be that Tiny Core wasn't built to run in cmd-line mode only. I could be wrong.

-::-Try holding down Ctrl + Alt +F1 and see if that gives you a cmd-line prompt?-::-

Did that laptop have an os on it when you first got it?

It didnt have an OS no. Well I just tried using both the arch linix boot usb and the tiny core one and neither loaded graphics so it wont be a software thing in that case will it?

It has 14 days return so I will try stuff the rest of the weekend and send it back if no succes.

Since it didn't have an os when you got it I hope who ever wiped the os that was on it didn't hose the BIOS.
Can you get into the BIOS?

No, it won't be a software thing.
I'd send it back too if you don't get it to work by the end of the weekend.

It could be that it doesn't have enough RAM (the RAM could be bad) or the graphics card may be on it's way out.

Try running Puppy Linux live. If that doesn't boot up with a GUI than I'd lean more toward thinking it's a GPU problem.
http://puppylinux.org/main/Download

Running lspci will show you the spec's of the laptop and the GPU as well.

I just connected it via cable to my main computer's monitor and it was botting fine and normal so the screen is the problem.

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Sounds like you have gotten to the root of the problem.
Evidently the screen is shot.

I'm not sure how much of an expense it would be to replace the screen.
Your local computer repair shop could give you a ball park idea; that is if you even want to mess with that.

Me; personally, I'd return that Dell to the merchant on e-bay and purchase another laptop.

Don't forget to mark your thread SOLVED:-