Can my laptops graphics card be upgraded?

No. Most laptops have no "graphics card". They have either an integrated Intel or AMD APU (a CPU with GPU on same chip) or they have a graphics chip soldered to the motherboard.
Only a few top end Dell or Asus or Sager or some other brands laptops have tiny "daughter boards" with a GPU chip on them.
If you bought that 625m chip, I don't know how you could make it work on your motherboard, as there's no spot for it. The power and cooling would be a problem, as your laptop has no extra.
Next time buy a gaming laptop ($$$$) a good desktop that can take many thousands of different cards.
As far as a "slot" in a laptop for a graphics card, there are only two I have seen:
Inside the computer is a PCI-e 1X connection for a wireless "card". If you use that with a cable, you can go out to an external box for power, cooling and external monitor. How you get the cable out through the laptop case is up to you. Warranty would be voided.
Or if the outside of the laptop has a 34mm or 50mm wide slot for Expresscard (few do), you can hook a cable to the external box. That box, the desktop graphics card you choose and a monitor would be at least $600.
I have heard of people using a USB 3.0 port with an external video docking station for fair results by DVI or HDMI with the external monitor.