How to Copy Files from Computer with Bad Onboard Video Card

I have a 3 year old HP Compaq and the onboard video card started burning and now i get no picture. I want to copy some files off the computer before the burning destoys the rest of the motherboard (i can hear start up sounds when i turn the pc on). How can i do this? Best Buy said they can turn off the onboard video card so i can put in a new one, but will charge $50 to do it. And of course even if that works i've got the problem of the motherboard burn spot slowly enlarging when the power is on. I know, i know, i should've back up the system more often

Playing around with a damaged motherboard risks frying the rest of the hardware, including the HDD electronics.
I think it would be best to either drop in a cheap replacement motherboard or take out the drive/s and copy the files over to a new system.

Any data stored in memory is lost when the computer is turned off-it is termed volatile memory-all your permanent storage is through the hard drive.
The drives are usually held in place by 4 small screws, two on each side, so you will need to take off both sides of the case to reach them, the drive should then slide out backwards, into the case but that last detail varies and in some cases (pun intended) you will need to take off the front to withdraw the drive that way.
You do not sound very knowlageable about the hardware. If you intend to replace the motherboard yourself I suggest you spend some time on Google researching the subject. Motherboard replacement is not that hard but there are points you should be aware of before proceeding.
If you do not feel confident I suggest you look through the phone book for a small local store that can do the job.
As for stupid questions: "The importaint thing is to keep questioning" A Eienstien. (not a verbatim quote but it is close enough).