Recover iphone backup from broken laptop?

I have my iphone backed up to my old laptop that the sreen no longer works on, it charges and lights up when plugged in but i get no response from the screen at all. I lost my iphone and had to get a new one, but need to access the backup that is on my old laptop, is there any way of extracting this data from my old laptop and putting it on my new one?

If the laptop still works, is there a place to plug in an external monitor? Then you can easily use the laptop like it was.
If it doesn't work at all, then you may be able to remove the hard drive and plug it into another working computer to pull the data off of it.

The data is on the old hard drive. It is possible for a savy guy to remove it and
place it in a usb adapter to access it on the new laptop.